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Known
df_sslt__df_scut__df_made__df_old__df_new__df_left__df_right__df_txp__df_pprod__df_sset__df_trans__df_bigcup__df_fix__df_limits__df_funs__df_singleton__df_singles__df_image
:
∀ x0 : ο .
(
wceq
csslt
(
copab
(
λ x1 x2 .
w3a
(
wss
(
cv
x1
)
csur
)
(
wss
(
cv
x2
)
csur
)
(
wral
(
λ x3 .
wral
(
λ x4 .
wbr
(
cv
x3
)
(
cv
x4
)
cslt
)
(
λ x4 .
cv
x2
)
)
(
λ x3 .
cv
x1
)
)
)
)
⟶
wceq
cscut
(
cmpt2
(
λ x1 x2 .
cpw
csur
)
(
λ x1 x2 .
cima
csslt
(
csn
(
cv
x1
)
)
)
(
λ x1 x2 .
crio
(
λ x3 .
wceq
(
cfv
(
cv
x3
)
cbday
)
(
cint
(
cima
cbday
(
crab
(
λ x4 .
wa
(
wbr
(
cv
x1
)
(
csn
(
cv
x4
)
)
csslt
)
(
wbr
(
csn
(
cv
x4
)
)
(
cv
x2
)
csslt
)
)
(
λ x4 .
csur
)
)
)
)
)
(
λ x3 .
crab
(
λ x4 .
wa
(
wbr
(
cv
x1
)
(
csn
(
cv
x4
)
)
csslt
)
(
wbr
(
csn
(
cv
x4
)
)
(
cv
x2
)
csslt
)
)
(
λ x4 .
csur
)
)
)
)
⟶
wceq
cmade
(
crecs
(
cmpt
(
λ x1 .
cvv
)
(
λ x1 .
cima
cscut
(
cxp
(
cpw
(
cuni
(
crn
(
cv
x1
)
)
)
)
(
cpw
(
cuni
(
crn
(
cv
x1
)
)
)
)
)
)
)
)
⟶
wceq
cold
(
cmpt
(
λ x1 .
con0
)
(
λ x1 .
cuni
(
cima
cmade
(
cv
x1
)
)
)
)
⟶
wceq
cnew
(
cmpt
(
λ x1 .
con0
)
(
λ x1 .
cdif
(
cfv
(
cv
x1
)
cold
)
(
cfv
(
cv
x1
)
cmade
)
)
)
⟶
wceq
cleft
(
cmpt
(
λ x1 .
csur
)
(
λ x1 .
crab
(
λ x2 .
wral
(
λ x3 .
wa
(
wbr
(
cv
x2
)
(
cv
x3
)
cslt
)
(
wbr
(
cv
x3
)
(
cv
x1
)
cslt
)
⟶
wcel
(
cfv
(
cv
x2
)
cbday
)
(
cfv
(
cv
x3
)
cbday
)
)
(
λ x3 .
csur
)
)
(
λ x2 .
cfv
(
cfv
(
cv
x1
)
cbday
)
cold
)
)
)
⟶
wceq
cright
(
cmpt
(
λ x1 .
csur
)
(
λ x1 .
crab
(
λ x2 .
wral
(
λ x3 .
wa
(
wbr
(
cv
x1
)
(
cv
x3
)
cslt
)
(
wbr
(
cv
x3
)
(
cv
x2
)
cslt
)
⟶
wcel
(
cfv
(
cv
x2
)
cbday
)
(
cfv
(
cv
x3
)
cbday
)
)
(
λ x3 .
csur
)
)
(
λ x2 .
cfv
(
cfv
(
cv
x1
)
cbday
)
cold
)
)
)
⟶
(
∀ x1 x2 :
ι → ο
.
wceq
(
ctxp
x1
x2
)
(
cin
(
ccom
(
ccnv
(
cres
c1st
(
cxp
cvv
cvv
)
)
)
x1
)
(
ccom
(
ccnv
(
cres
c2nd
(
cxp
cvv
cvv
)
)
)
x2
)
)
)
⟶
(
∀ x1 x2 :
ι → ο
.
wceq
(
cpprod
x1
x2
)
(
ctxp
(
ccom
x1
(
cres
c1st
(
cxp
cvv
cvv
)
)
)
(
ccom
x2
(
cres
c2nd
(
cxp
cvv
cvv
)
)
)
)
)
⟶
wceq
csset
(
cdif
(
cxp
cvv
cvv
)
(
crn
(
ctxp
cep
(
cdif
cvv
cep
)
)
)
)
⟶
wceq
ctrans
(
cdif
cvv
(
crn
(
cdif
(
ccom
cep
cep
)
cep
)
)
)
⟶
wceq
cbigcup
(
cdif
(
cxp
cvv
cvv
)
(
crn
(
csymdif
(
ctxp
cvv
cep
)
(
ctxp
(
ccom
cep
cep
)
cvv
)
)
)
)
⟶
(
∀ x1 :
ι → ο
.
wceq
(
cfix
x1
)
(
cdm
(
cin
x1
cid
)
)
)
⟶
wceq
climits
(
cdif
(
cin
con0
(
cfix
cbigcup
)
)
(
csn
c0
)
)
⟶
wceq
cfuns
(
cdif
(
cpw
(
cxp
cvv
cvv
)
)
(
cfix
(
ccom
cep
(
ccom
(
ctxp
c1st
(
ccom
(
cdif
cvv
cid
)
c2nd
)
)
(
ccnv
cep
)
)
)
)
)
⟶
wceq
csingle
(
cdif
(
cxp
cvv
cvv
)
(
crn
(
csymdif
(
ctxp
cvv
cep
)
(
ctxp
cid
cvv
)
)
)
)
⟶
wceq
csingles
(
crn
csingle
)
⟶
(
∀ x1 :
ι → ο
.
wceq
(
cimage
x1
)
(
cdif
(
cxp
cvv
cvv
)
(
crn
(
csymdif
(
ctxp
cvv
cep
)
(
ctxp
(
ccom
cep
(
ccnv
x1
)
)
cvv
)
)
)
)
)
⟶
x0
)
⟶
x0
Theorem
df_sslt
:
wceq
csslt
(
copab
(
λ x0 x1 .
w3a
(
wss
(
cv
x0
)
csur
)
(
wss
(
cv
x1
)
csur
)
(
wral
(
λ x2 .
wral
(
λ x3 .
wbr
(
cv
x2
)
(
cv
x3
)
cslt
)
(
λ x3 .
cv
x1
)
)
(
λ x2 .
cv
x0
)
)
)
)
(proof)
Theorem
df_scut
:
wceq
cscut
(
cmpt2
(
λ x0 x1 .
cpw
csur
)
(
λ x0 x1 .
cima
csslt
(
csn
(
cv
x0
)
)
)
(
λ x0 x1 .
crio
(
λ x2 .
wceq
(
cfv
(
cv
x2
)
cbday
)
(
cint
(
cima
cbday
(
crab
(
λ x3 .
wa
(
wbr
(
cv
x0
)
(
csn
(
cv
x3
)
)
csslt
)
(
wbr
(
csn
(
cv
x3
)
)
(
cv
x1
)
csslt
)
)
(
λ x3 .
csur
)
)
)
)
)
(
λ x2 .
crab
(
λ x3 .
wa
(
wbr
(
cv
x0
)
(
csn
(
cv
x3
)
)
csslt
)
(
wbr
(
csn
(
cv
x3
)
)
(
cv
x1
)
csslt
)
)
(
λ x3 .
csur
)
)
)
)
(proof)
Theorem
df_made
:
wceq
cmade
(
crecs
(
cmpt
(
λ x0 .
cvv
)
(
λ x0 .
cima
cscut
(
cxp
(
cpw
(
cuni
(
crn
(
cv
x0
)
)
)
)
(
cpw
(
cuni
(
crn
(
cv
x0
)
)
)
)
)
)
)
)
(proof)
Theorem
df_old
:
wceq
cold
(
cmpt
(
λ x0 .
con0
)
(
λ x0 .
cuni
(
cima
cmade
(
cv
x0
)
)
)
)
(proof)
Theorem
df_new
:
wceq
cnew
(
cmpt
(
λ x0 .
con0
)
(
λ x0 .
cdif
(
cfv
(
cv
x0
)
cold
)
(
cfv
(
cv
x0
)
cmade
)
)
)
(proof)
Theorem
df_left
:
wceq
cleft
(
cmpt
(
λ x0 .
csur
)
(
λ x0 .
crab
(
λ x1 .
wral
(
λ x2 .
wa
(
wbr
(
cv
x1
)
(
cv
x2
)
cslt
)
(
wbr
(
cv
x2
)
(
cv
x0
)
cslt
)
⟶
wcel
(
cfv
(
cv
x1
)
cbday
)
(
cfv
(
cv
x2
)
cbday
)
)
(
λ x2 .
csur
)
)
(
λ x1 .
cfv
(
cfv
(
cv
x0
)
cbday
)
cold
)
)
)
(proof)
Theorem
df_right
:
wceq
cright
(
cmpt
(
λ x0 .
csur
)
(
λ x0 .
crab
(
λ x1 .
wral
(
λ x2 .
wa
(
wbr
(
cv
x0
)
(
cv
x2
)
cslt
)
(
wbr
(
cv
x2
)
(
cv
x1
)
cslt
)
⟶
wcel
(
cfv
(
cv
x1
)
cbday
)
(
cfv
(
cv
x2
)
cbday
)
)
(
λ x2 .
csur
)
)
(
λ x1 .
cfv
(
cfv
(
cv
x0
)
cbday
)
cold
)
)
)
(proof)
Theorem
df_txp
:
∀ x0 x1 :
ι → ο
.
wceq
(
ctxp
x0
x1
)
(
cin
(
ccom
(
ccnv
(
cres
c1st
(
cxp
cvv
cvv
)
)
)
x0
)
(
ccom
(
ccnv
(
cres
c2nd
(
cxp
cvv
cvv
)
)
)
x1
)
)
(proof)
Theorem
df_pprod
:
∀ x0 x1 :
ι → ο
.
wceq
(
cpprod
x0
x1
)
(
ctxp
(
ccom
x0
(
cres
c1st
(
cxp
cvv
cvv
)
)
)
(
ccom
x1
(
cres
c2nd
(
cxp
cvv
cvv
)
)
)
)
(proof)
Theorem
df_sset
:
wceq
csset
(
cdif
(
cxp
cvv
cvv
)
(
crn
(
ctxp
cep
(
cdif
cvv
cep
)
)
)
)
(proof)
Theorem
df_trans
:
wceq
ctrans
(
cdif
cvv
(
crn
(
cdif
(
ccom
cep
cep
)
cep
)
)
)
(proof)
Theorem
df_bigcup
:
wceq
cbigcup
(
cdif
(
cxp
cvv
cvv
)
(
crn
(
csymdif
(
ctxp
cvv
cep
)
(
ctxp
(
ccom
cep
cep
)
cvv
)
)
)
)
(proof)
Theorem
df_fix
:
∀ x0 :
ι → ο
.
wceq
(
cfix
x0
)
(
cdm
(
cin
x0
cid
)
)
(proof)
Theorem
df_limits
:
wceq
climits
(
cdif
(
cin
con0
(
cfix
cbigcup
)
)
(
csn
c0
)
)
(proof)
Theorem
df_funs
:
wceq
cfuns
(
cdif
(
cpw
(
cxp
cvv
cvv
)
)
(
cfix
(
ccom
cep
(
ccom
(
ctxp
c1st
(
ccom
(
cdif
cvv
cid
)
c2nd
)
)
(
ccnv
cep
)
)
)
)
)
(proof)
Theorem
df_singleton
:
wceq
csingle
(
cdif
(
cxp
cvv
cvv
)
(
crn
(
csymdif
(
ctxp
cvv
cep
)
(
ctxp
cid
cvv
)
)
)
)
(proof)
Theorem
df_singles
:
wceq
csingles
(
crn
csingle
)
(proof)
Theorem
df_image
:
∀ x0 :
ι → ο
.
wceq
(
cimage
x0
)
(
cdif
(
cxp
cvv
cvv
)
(
crn
(
csymdif
(
ctxp
cvv
cep
)
(
ctxp
(
ccom
cep
(
ccnv
x0
)
)
cvv
)
)
)
)
(proof)