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Theorem catciso 15953
Description: A functor is an isomorphism of categories if and only if it is full and faithful, and is a bijection on the objects. Remark 3.28(2) in [Adamek] p. 34. (Contributed by Mario Carneiro, 29-Jan-2017.)
Hypotheses
Ref Expression
catciso.c  |-  C  =  (CatCat `  U )
catciso.b  |-  B  =  ( Base `  C
)
catciso.r  |-  R  =  ( Base `  X
)
catciso.s  |-  S  =  ( Base `  Y
)
catciso.u  |-  ( ph  ->  U  e.  V )
catciso.x  |-  ( ph  ->  X  e.  B )
catciso.y  |-  ( ph  ->  Y  e.  B )
catciso.i  |-  I  =  (  Iso  `  C
)
Assertion
Ref Expression
catciso  |-  ( ph  ->  ( F  e.  ( X I Y )  <-> 
( F  e.  ( ( X Full  Y )  i^i  ( X Faith  Y
) )  /\  ( 1st `  F ) : R -1-1-onto-> S ) ) )

Proof of Theorem catciso
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relfunc 15718 . . . . 5  |-  Rel  ( X  Func  Y )
2 catciso.b . . . . . . . . . . . . . 14  |-  B  =  ( Base `  C
)
3 eqid 2429 . . . . . . . . . . . . . 14  |-  (Inv `  C )  =  (Inv
`  C )
4 catciso.u . . . . . . . . . . . . . . 15  |-  ( ph  ->  U  e.  V )
5 catciso.c . . . . . . . . . . . . . . . 16  |-  C  =  (CatCat `  U )
65catccat 15950 . . . . . . . . . . . . . . 15  |-  ( U  e.  V  ->  C  e.  Cat )
74, 6syl 17 . . . . . . . . . . . . . 14  |-  ( ph  ->  C  e.  Cat )
8 catciso.x . . . . . . . . . . . . . 14  |-  ( ph  ->  X  e.  B )
9 catciso.y . . . . . . . . . . . . . 14  |-  ( ph  ->  Y  e.  B )
10 catciso.i . . . . . . . . . . . . . 14  |-  I  =  (  Iso  `  C
)
112, 3, 7, 8, 9, 10isoval 15621 . . . . . . . . . . . . 13  |-  ( ph  ->  ( X I Y )  =  dom  ( X (Inv `  C ) Y ) )
1211eleq2d 2499 . . . . . . . . . . . 12  |-  ( ph  ->  ( F  e.  ( X I Y )  <-> 
F  e.  dom  ( X (Inv `  C ) Y ) ) )
1312biimpa 486 . . . . . . . . . . 11  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  F  e.  dom  ( X (Inv `  C ) Y ) )
147adantr 466 . . . . . . . . . . . . 13  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  C  e.  Cat )
158adantr 466 . . . . . . . . . . . . 13  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  X  e.  B )
169adantr 466 . . . . . . . . . . . . 13  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  Y  e.  B )
172, 3, 14, 15, 16invfun 15620 . . . . . . . . . . . 12  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  Fun  ( X (Inv `  C ) Y ) )
18 funfvbrb 6010 . . . . . . . . . . . 12  |-  ( Fun  ( X (Inv `  C ) Y )  ->  ( F  e. 
dom  ( X (Inv
`  C ) Y )  <->  F ( X (Inv
`  C ) Y ) ( ( X (Inv `  C ) Y ) `  F
) ) )
1917, 18syl 17 . . . . . . . . . . 11  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  ( F  e.  dom  ( X (Inv
`  C ) Y )  <->  F ( X (Inv
`  C ) Y ) ( ( X (Inv `  C ) Y ) `  F
) ) )
2013, 19mpbid 213 . . . . . . . . . 10  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  F ( X (Inv `  C ) Y ) ( ( X (Inv `  C
) Y ) `  F ) )
21 eqid 2429 . . . . . . . . . . 11  |-  (Sect `  C )  =  (Sect `  C )
222, 3, 14, 15, 16, 21isinv 15616 . . . . . . . . . 10  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  ( F
( X (Inv `  C ) Y ) ( ( X (Inv
`  C ) Y ) `  F )  <-> 
( F ( X (Sect `  C ) Y ) ( ( X (Inv `  C
) Y ) `  F )  /\  (
( X (Inv `  C ) Y ) `
 F ) ( Y (Sect `  C
) X ) F ) ) )
2320, 22mpbid 213 . . . . . . . . 9  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  ( F
( X (Sect `  C ) Y ) ( ( X (Inv
`  C ) Y ) `  F )  /\  ( ( X (Inv `  C ) Y ) `  F
) ( Y (Sect `  C ) X ) F ) )
2423simpld 460 . . . . . . . 8  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  F ( X (Sect `  C ) Y ) ( ( X (Inv `  C
) Y ) `  F ) )
25 eqid 2429 . . . . . . . . 9  |-  ( Hom  `  C )  =  ( Hom  `  C )
26 eqid 2429 . . . . . . . . 9  |-  (comp `  C )  =  (comp `  C )
27 eqid 2429 . . . . . . . . 9  |-  ( Id
`  C )  =  ( Id `  C
)
282, 25, 26, 27, 21, 14, 15, 16issect 15609 . . . . . . . 8  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  ( F
( X (Sect `  C ) Y ) ( ( X (Inv
`  C ) Y ) `  F )  <-> 
( F  e.  ( X ( Hom  `  C
) Y )  /\  ( ( X (Inv
`  C ) Y ) `  F )  e.  ( Y ( Hom  `  C ) X )  /\  (
( ( X (Inv
`  C ) Y ) `  F ) ( <. X ,  Y >. (comp `  C ) X ) F )  =  ( ( Id
`  C ) `  X ) ) ) )
2924, 28mpbid 213 . . . . . . 7  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  ( F  e.  ( X ( Hom  `  C ) Y )  /\  ( ( X (Inv `  C ) Y ) `  F
)  e.  ( Y ( Hom  `  C
) X )  /\  ( ( ( X (Inv `  C ) Y ) `  F
) ( <. X ,  Y >. (comp `  C
) X ) F )  =  ( ( Id `  C ) `
 X ) ) )
3029simp1d 1017 . . . . . 6  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  F  e.  ( X ( Hom  `  C
) Y ) )
315, 2, 4, 25, 8, 9catchom 15945 . . . . . . 7  |-  ( ph  ->  ( X ( Hom  `  C ) Y )  =  ( X  Func  Y ) )
3231adantr 466 . . . . . 6  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  ( X
( Hom  `  C ) Y )  =  ( X  Func  Y )
)
3330, 32eleqtrd 2519 . . . . 5  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  F  e.  ( X  Func  Y ) )
34 1st2nd 6853 . . . . 5  |-  ( ( Rel  ( X  Func  Y )  /\  F  e.  ( X  Func  Y
) )  ->  F  =  <. ( 1st `  F
) ,  ( 2nd `  F ) >. )
351, 33, 34sylancr 667 . . . 4  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  F  =  <. ( 1st `  F
) ,  ( 2nd `  F ) >. )
36 1st2ndbr 6856 . . . . . . 7  |-  ( ( Rel  ( X  Func  Y )  /\  F  e.  ( X  Func  Y
) )  ->  ( 1st `  F ) ( X  Func  Y )
( 2nd `  F
) )
371, 33, 36sylancr 667 . . . . . 6  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  ( 1st `  F ) ( X 
Func  Y ) ( 2nd `  F ) )
38 catciso.r . . . . . . . . 9  |-  R  =  ( Base `  X
)
39 eqid 2429 . . . . . . . . 9  |-  ( Hom  `  X )  =  ( Hom  `  X )
40 eqid 2429 . . . . . . . . 9  |-  ( Hom  `  Y )  =  ( Hom  `  Y )
4137adantr 466 . . . . . . . . 9  |-  ( ( ( ph  /\  F  e.  ( X I Y ) )  /\  (
x  e.  R  /\  y  e.  R )
)  ->  ( 1st `  F ) ( X 
Func  Y ) ( 2nd `  F ) )
42 simprl 762 . . . . . . . . 9  |-  ( ( ( ph  /\  F  e.  ( X I Y ) )  /\  (
x  e.  R  /\  y  e.  R )
)  ->  x  e.  R )
43 simprr 764 . . . . . . . . 9  |-  ( ( ( ph  /\  F  e.  ( X I Y ) )  /\  (
x  e.  R  /\  y  e.  R )
)  ->  y  e.  R )
4438, 39, 40, 41, 42, 43funcf2 15724 . . . . . . . 8  |-  ( ( ( ph  /\  F  e.  ( X I Y ) )  /\  (
x  e.  R  /\  y  e.  R )
)  ->  ( x
( 2nd `  F
) y ) : ( x ( Hom  `  X ) y ) --> ( ( ( 1st `  F ) `  x
) ( Hom  `  Y
) ( ( 1st `  F ) `  y
) ) )
45 catciso.s . . . . . . . . . 10  |-  S  =  ( Base `  Y
)
46 relfunc 15718 . . . . . . . . . . . 12  |-  Rel  ( Y  Func  X )
4729simp2d 1018 . . . . . . . . . . . . 13  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  ( ( X (Inv `  C ) Y ) `  F
)  e.  ( Y ( Hom  `  C
) X ) )
485, 2, 4, 25, 9, 8catchom 15945 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( Y ( Hom  `  C ) X )  =  ( Y  Func  X ) )
4948adantr 466 . . . . . . . . . . . . 13  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  ( Y
( Hom  `  C ) X )  =  ( Y  Func  X )
)
5047, 49eleqtrd 2519 . . . . . . . . . . . 12  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  ( ( X (Inv `  C ) Y ) `  F
)  e.  ( Y 
Func  X ) )
51 1st2ndbr 6856 . . . . . . . . . . . 12  |-  ( ( Rel  ( Y  Func  X )  /\  ( ( X (Inv `  C
) Y ) `  F )  e.  ( Y  Func  X )
)  ->  ( 1st `  ( ( X (Inv
`  C ) Y ) `  F ) ) ( Y  Func  X ) ( 2nd `  (
( X (Inv `  C ) Y ) `
 F ) ) )
5246, 50, 51sylancr 667 . . . . . . . . . . 11  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  ( 1st `  ( ( X (Inv
`  C ) Y ) `  F ) ) ( Y  Func  X ) ( 2nd `  (
( X (Inv `  C ) Y ) `
 F ) ) )
5352adantr 466 . . . . . . . . . 10  |-  ( ( ( ph  /\  F  e.  ( X I Y ) )  /\  (
x  e.  R  /\  y  e.  R )
)  ->  ( 1st `  ( ( X (Inv
`  C ) Y ) `  F ) ) ( Y  Func  X ) ( 2nd `  (
( X (Inv `  C ) Y ) `
 F ) ) )
5438, 45, 41funcf1 15722 . . . . . . . . . . 11  |-  ( ( ( ph  /\  F  e.  ( X I Y ) )  /\  (
x  e.  R  /\  y  e.  R )
)  ->  ( 1st `  F ) : R --> S )
5554, 42ffvelrnd 6038 . . . . . . . . . 10  |-  ( ( ( ph  /\  F  e.  ( X I Y ) )  /\  (
x  e.  R  /\  y  e.  R )
)  ->  ( ( 1st `  F ) `  x )  e.  S
)
5654, 43ffvelrnd 6038 . . . . . . . . . 10  |-  ( ( ( ph  /\  F  e.  ( X I Y ) )  /\  (
x  e.  R  /\  y  e.  R )
)  ->  ( ( 1st `  F ) `  y )  e.  S
)
5745, 40, 39, 53, 55, 56funcf2 15724 . . . . . . . . 9  |-  ( ( ( ph  /\  F  e.  ( X I Y ) )  /\  (
x  e.  R  /\  y  e.  R )
)  ->  ( (
( 1st `  F
) `  x )
( 2nd `  (
( X (Inv `  C ) Y ) `
 F ) ) ( ( 1st `  F
) `  y )
) : ( ( ( 1st `  F
) `  x )
( Hom  `  Y ) ( ( 1st `  F
) `  y )
) --> ( ( ( 1st `  ( ( X (Inv `  C
) Y ) `  F ) ) `  ( ( 1st `  F
) `  x )
) ( Hom  `  X
) ( ( 1st `  ( ( X (Inv
`  C ) Y ) `  F ) ) `  ( ( 1st `  F ) `
 y ) ) ) )
5829simp3d 1019 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  ( (
( X (Inv `  C ) Y ) `
 F ) (
<. X ,  Y >. (comp `  C ) X ) F )  =  ( ( Id `  C
) `  X )
)
594adantr 466 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  U  e.  V )
605, 2, 59, 26, 15, 16, 15, 33, 50catcco 15947 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  ( (
( X (Inv `  C ) Y ) `
 F ) (
<. X ,  Y >. (comp `  C ) X ) F )  =  ( ( ( X (Inv
`  C ) Y ) `  F )  o.func 
F ) )
61 eqid 2429 . . . . . . . . . . . . . . . . . 18  |-  (idfunc `  X
)  =  (idfunc `  X
)
625, 2, 27, 61, 4, 8catcid 15949 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( ( Id `  C ) `  X
)  =  (idfunc `  X
) )
6362adantr 466 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  ( ( Id `  C ) `  X )  =  (idfunc `  X ) )
6458, 60, 633eqtr3d 2478 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  ( (
( X (Inv `  C ) Y ) `
 F )  o.func  F
)  =  (idfunc `  X
) )
6564adantr 466 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  F  e.  ( X I Y ) )  /\  (
x  e.  R  /\  y  e.  R )
)  ->  ( (
( X (Inv `  C ) Y ) `
 F )  o.func  F
)  =  (idfunc `  X
) )
6665fveq2d 5885 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  F  e.  ( X I Y ) )  /\  (
x  e.  R  /\  y  e.  R )
)  ->  ( 1st `  ( ( ( X (Inv `  C ) Y ) `  F
)  o.func 
F ) )  =  ( 1st `  (idfunc `  X
) ) )
6766fveq1d 5883 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  F  e.  ( X I Y ) )  /\  (
x  e.  R  /\  y  e.  R )
)  ->  ( ( 1st `  ( ( ( X (Inv `  C
) Y ) `  F )  o.func  F )
) `  x )  =  ( ( 1st `  (idfunc `  X ) ) `  x ) )
6833adantr 466 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  F  e.  ( X I Y ) )  /\  (
x  e.  R  /\  y  e.  R )
)  ->  F  e.  ( X  Func  Y ) )
6950adantr 466 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  F  e.  ( X I Y ) )  /\  (
x  e.  R  /\  y  e.  R )
)  ->  ( ( X (Inv `  C ) Y ) `  F
)  e.  ( Y 
Func  X ) )
7038, 68, 69, 42cofu1 15740 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  F  e.  ( X I Y ) )  /\  (
x  e.  R  /\  y  e.  R )
)  ->  ( ( 1st `  ( ( ( X (Inv `  C
) Y ) `  F )  o.func  F )
) `  x )  =  ( ( 1st `  ( ( X (Inv
`  C ) Y ) `  F ) ) `  ( ( 1st `  F ) `
 x ) ) )
715, 2, 4catcbas 15943 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  B  =  ( U  i^i  Cat ) )
72 inss2 3689 . . . . . . . . . . . . . . . 16  |-  ( U  i^i  Cat )  C_  Cat
7371, 72syl6eqss 3520 . . . . . . . . . . . . . . 15  |-  ( ph  ->  B  C_  Cat )
7473, 8sseldd 3471 . . . . . . . . . . . . . 14  |-  ( ph  ->  X  e.  Cat )
7574ad2antrr 730 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  F  e.  ( X I Y ) )  /\  (
x  e.  R  /\  y  e.  R )
)  ->  X  e.  Cat )
7661, 38, 75, 42idfu1 15736 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  F  e.  ( X I Y ) )  /\  (
x  e.  R  /\  y  e.  R )
)  ->  ( ( 1st `  (idfunc `  X ) ) `  x )  =  x )
7767, 70, 763eqtr3d 2478 . . . . . . . . . . 11  |-  ( ( ( ph  /\  F  e.  ( X I Y ) )  /\  (
x  e.  R  /\  y  e.  R )
)  ->  ( ( 1st `  ( ( X (Inv `  C ) Y ) `  F
) ) `  (
( 1st `  F
) `  x )
)  =  x )
7866fveq1d 5883 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  F  e.  ( X I Y ) )  /\  (
x  e.  R  /\  y  e.  R )
)  ->  ( ( 1st `  ( ( ( X (Inv `  C
) Y ) `  F )  o.func  F )
) `  y )  =  ( ( 1st `  (idfunc `  X ) ) `  y ) )
7938, 68, 69, 43cofu1 15740 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  F  e.  ( X I Y ) )  /\  (
x  e.  R  /\  y  e.  R )
)  ->  ( ( 1st `  ( ( ( X (Inv `  C
) Y ) `  F )  o.func  F )
) `  y )  =  ( ( 1st `  ( ( X (Inv
`  C ) Y ) `  F ) ) `  ( ( 1st `  F ) `
 y ) ) )
8061, 38, 75, 43idfu1 15736 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  F  e.  ( X I Y ) )  /\  (
x  e.  R  /\  y  e.  R )
)  ->  ( ( 1st `  (idfunc `  X ) ) `  y )  =  y )
8178, 79, 803eqtr3d 2478 . . . . . . . . . . 11  |-  ( ( ( ph  /\  F  e.  ( X I Y ) )  /\  (
x  e.  R  /\  y  e.  R )
)  ->  ( ( 1st `  ( ( X (Inv `  C ) Y ) `  F
) ) `  (
( 1st `  F
) `  y )
)  =  y )
8277, 81oveq12d 6323 . . . . . . . . . 10  |-  ( ( ( ph  /\  F  e.  ( X I Y ) )  /\  (
x  e.  R  /\  y  e.  R )
)  ->  ( (
( 1st `  (
( X (Inv `  C ) Y ) `
 F ) ) `
 ( ( 1st `  F ) `  x
) ) ( Hom  `  X ) ( ( 1st `  ( ( X (Inv `  C
) Y ) `  F ) ) `  ( ( 1st `  F
) `  y )
) )  =  ( x ( Hom  `  X
) y ) )
8382feq3d 5734 . . . . . . . . 9  |-  ( ( ( ph  /\  F  e.  ( X I Y ) )  /\  (
x  e.  R  /\  y  e.  R )
)  ->  ( (
( ( 1st `  F
) `  x )
( 2nd `  (
( X (Inv `  C ) Y ) `
 F ) ) ( ( 1st `  F
) `  y )
) : ( ( ( 1st `  F
) `  x )
( Hom  `  Y ) ( ( 1st `  F
) `  y )
) --> ( ( ( 1st `  ( ( X (Inv `  C
) Y ) `  F ) ) `  ( ( 1st `  F
) `  x )
) ( Hom  `  X
) ( ( 1st `  ( ( X (Inv
`  C ) Y ) `  F ) ) `  ( ( 1st `  F ) `
 y ) ) )  <->  ( ( ( 1st `  F ) `
 x ) ( 2nd `  ( ( X (Inv `  C
) Y ) `  F ) ) ( ( 1st `  F
) `  y )
) : ( ( ( 1st `  F
) `  x )
( Hom  `  Y ) ( ( 1st `  F
) `  y )
) --> ( x ( Hom  `  X )
y ) ) )
8457, 83mpbid 213 . . . . . . . 8  |-  ( ( ( ph  /\  F  e.  ( X I Y ) )  /\  (
x  e.  R  /\  y  e.  R )
)  ->  ( (
( 1st `  F
) `  x )
( 2nd `  (
( X (Inv `  C ) Y ) `
 F ) ) ( ( 1st `  F
) `  y )
) : ( ( ( 1st `  F
) `  x )
( Hom  `  Y ) ( ( 1st `  F
) `  y )
) --> ( x ( Hom  `  X )
y ) )
8565fveq2d 5885 . . . . . . . . . 10  |-  ( ( ( ph  /\  F  e.  ( X I Y ) )  /\  (
x  e.  R  /\  y  e.  R )
)  ->  ( 2nd `  ( ( ( X (Inv `  C ) Y ) `  F
)  o.func 
F ) )  =  ( 2nd `  (idfunc `  X
) ) )
8685oveqd 6322 . . . . . . . . 9  |-  ( ( ( ph  /\  F  e.  ( X I Y ) )  /\  (
x  e.  R  /\  y  e.  R )
)  ->  ( x
( 2nd `  (
( ( X (Inv
`  C ) Y ) `  F )  o.func 
F ) ) y )  =  ( x ( 2nd `  (idfunc `  X
) ) y ) )
8738, 68, 69, 42, 43cofu2nd 15741 . . . . . . . . 9  |-  ( ( ( ph  /\  F  e.  ( X I Y ) )  /\  (
x  e.  R  /\  y  e.  R )
)  ->  ( x
( 2nd `  (
( ( X (Inv
`  C ) Y ) `  F )  o.func 
F ) ) y )  =  ( ( ( ( 1st `  F
) `  x )
( 2nd `  (
( X (Inv `  C ) Y ) `
 F ) ) ( ( 1st `  F
) `  y )
)  o.  ( x ( 2nd `  F
) y ) ) )
8861, 38, 75, 39, 42, 43idfu2nd 15733 . . . . . . . . 9  |-  ( ( ( ph  /\  F  e.  ( X I Y ) )  /\  (
x  e.  R  /\  y  e.  R )
)  ->  ( x
( 2nd `  (idfunc `  X
) ) y )  =  (  _I  |`  (
x ( Hom  `  X
) y ) ) )
8986, 87, 883eqtr3d 2478 . . . . . . . 8  |-  ( ( ( ph  /\  F  e.  ( X I Y ) )  /\  (
x  e.  R  /\  y  e.  R )
)  ->  ( (
( ( 1st `  F
) `  x )
( 2nd `  (
( X (Inv `  C ) Y ) `
 F ) ) ( ( 1st `  F
) `  y )
)  o.  ( x ( 2nd `  F
) y ) )  =  (  _I  |`  (
x ( Hom  `  X
) y ) ) )
9023simprd 464 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  ( ( X (Inv `  C ) Y ) `  F
) ( Y (Sect `  C ) X ) F )
912, 25, 26, 27, 21, 14, 16, 15issect 15609 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  ( (
( X (Inv `  C ) Y ) `
 F ) ( Y (Sect `  C
) X ) F  <-> 
( ( ( X (Inv `  C ) Y ) `  F
)  e.  ( Y ( Hom  `  C
) X )  /\  F  e.  ( X
( Hom  `  C ) Y )  /\  ( F ( <. Y ,  X >. (comp `  C
) Y ) ( ( X (Inv `  C ) Y ) `
 F ) )  =  ( ( Id
`  C ) `  Y ) ) ) )
9290, 91mpbid 213 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  ( (
( X (Inv `  C ) Y ) `
 F )  e.  ( Y ( Hom  `  C ) X )  /\  F  e.  ( X ( Hom  `  C
) Y )  /\  ( F ( <. Y ,  X >. (comp `  C
) Y ) ( ( X (Inv `  C ) Y ) `
 F ) )  =  ( ( Id
`  C ) `  Y ) ) )
9392simp3d 1019 . . . . . . . . . . . . 13  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  ( F
( <. Y ,  X >. (comp `  C ) Y ) ( ( X (Inv `  C
) Y ) `  F ) )  =  ( ( Id `  C ) `  Y
) )
945, 2, 59, 26, 16, 15, 16, 50, 33catcco 15947 . . . . . . . . . . . . 13  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  ( F
( <. Y ,  X >. (comp `  C ) Y ) ( ( X (Inv `  C
) Y ) `  F ) )  =  ( F  o.func  ( ( X (Inv `  C ) Y ) `  F
) ) )
95 eqid 2429 . . . . . . . . . . . . . . 15  |-  (idfunc `  Y
)  =  (idfunc `  Y
)
965, 2, 27, 95, 4, 9catcid 15949 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( ( Id `  C ) `  Y
)  =  (idfunc `  Y
) )
9796adantr 466 . . . . . . . . . . . . 13  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  ( ( Id `  C ) `  Y )  =  (idfunc `  Y ) )
9893, 94, 973eqtr3d 2478 . . . . . . . . . . . 12  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  ( F  o.func  ( ( X (Inv
`  C ) Y ) `  F ) )  =  (idfunc `  Y
) )
9998adantr 466 . . . . . . . . . . 11  |-  ( ( ( ph  /\  F  e.  ( X I Y ) )  /\  (
x  e.  R  /\  y  e.  R )
)  ->  ( F  o.func  ( ( X (Inv
`  C ) Y ) `  F ) )  =  (idfunc `  Y
) )
10099fveq2d 5885 . . . . . . . . . 10  |-  ( ( ( ph  /\  F  e.  ( X I Y ) )  /\  (
x  e.  R  /\  y  e.  R )
)  ->  ( 2nd `  ( F  o.func  ( ( X (Inv `  C ) Y ) `  F
) ) )  =  ( 2nd `  (idfunc `  Y
) ) )
101100oveqd 6322 . . . . . . . . 9  |-  ( ( ( ph  /\  F  e.  ( X I Y ) )  /\  (
x  e.  R  /\  y  e.  R )
)  ->  ( (
( 1st `  F
) `  x )
( 2nd `  ( F  o.func  ( ( X (Inv
`  C ) Y ) `  F ) ) ) ( ( 1st `  F ) `
 y ) )  =  ( ( ( 1st `  F ) `
 x ) ( 2nd `  (idfunc `  Y
) ) ( ( 1st `  F ) `
 y ) ) )
10245, 69, 68, 55, 56cofu2nd 15741 . . . . . . . . . 10  |-  ( ( ( ph  /\  F  e.  ( X I Y ) )  /\  (
x  e.  R  /\  y  e.  R )
)  ->  ( (
( 1st `  F
) `  x )
( 2nd `  ( F  o.func  ( ( X (Inv
`  C ) Y ) `  F ) ) ) ( ( 1st `  F ) `
 y ) )  =  ( ( ( ( 1st `  (
( X (Inv `  C ) Y ) `
 F ) ) `
 ( ( 1st `  F ) `  x
) ) ( 2nd `  F ) ( ( 1st `  ( ( X (Inv `  C
) Y ) `  F ) ) `  ( ( 1st `  F
) `  y )
) )  o.  (
( ( 1st `  F
) `  x )
( 2nd `  (
( X (Inv `  C ) Y ) `
 F ) ) ( ( 1st `  F
) `  y )
) ) )
10377, 81oveq12d 6323 . . . . . . . . . . 11  |-  ( ( ( ph  /\  F  e.  ( X I Y ) )  /\  (
x  e.  R  /\  y  e.  R )
)  ->  ( (
( 1st `  (
( X (Inv `  C ) Y ) `
 F ) ) `
 ( ( 1st `  F ) `  x
) ) ( 2nd `  F ) ( ( 1st `  ( ( X (Inv `  C
) Y ) `  F ) ) `  ( ( 1st `  F
) `  y )
) )  =  ( x ( 2nd `  F
) y ) )
104103coeq1d 5016 . . . . . . . . . 10  |-  ( ( ( ph  /\  F  e.  ( X I Y ) )  /\  (
x  e.  R  /\  y  e.  R )
)  ->  ( (
( ( 1st `  (
( X (Inv `  C ) Y ) `
 F ) ) `
 ( ( 1st `  F ) `  x
) ) ( 2nd `  F ) ( ( 1st `  ( ( X (Inv `  C
) Y ) `  F ) ) `  ( ( 1st `  F
) `  y )
) )  o.  (
( ( 1st `  F
) `  x )
( 2nd `  (
( X (Inv `  C ) Y ) `
 F ) ) ( ( 1st `  F
) `  y )
) )  =  ( ( x ( 2nd `  F ) y )  o.  ( ( ( 1st `  F ) `
 x ) ( 2nd `  ( ( X (Inv `  C
) Y ) `  F ) ) ( ( 1st `  F
) `  y )
) ) )
105102, 104eqtrd 2470 . . . . . . . . 9  |-  ( ( ( ph  /\  F  e.  ( X I Y ) )  /\  (
x  e.  R  /\  y  e.  R )
)  ->  ( (
( 1st `  F
) `  x )
( 2nd `  ( F  o.func  ( ( X (Inv
`  C ) Y ) `  F ) ) ) ( ( 1st `  F ) `
 y ) )  =  ( ( x ( 2nd `  F
) y )  o.  ( ( ( 1st `  F ) `  x
) ( 2nd `  (
( X (Inv `  C ) Y ) `
 F ) ) ( ( 1st `  F
) `  y )
) ) )
10673ad2antrr 730 . . . . . . . . . . 11  |-  ( ( ( ph  /\  F  e.  ( X I Y ) )  /\  (
x  e.  R  /\  y  e.  R )
)  ->  B  C_  Cat )
1079ad2antrr 730 . . . . . . . . . . 11  |-  ( ( ( ph  /\  F  e.  ( X I Y ) )  /\  (
x  e.  R  /\  y  e.  R )
)  ->  Y  e.  B )
108106, 107sseldd 3471 . . . . . . . . . 10  |-  ( ( ( ph  /\  F  e.  ( X I Y ) )  /\  (
x  e.  R  /\  y  e.  R )
)  ->  Y  e.  Cat )
10995, 45, 108, 40, 55, 56idfu2nd 15733 . . . . . . . . 9  |-  ( ( ( ph  /\  F  e.  ( X I Y ) )  /\  (
x  e.  R  /\  y  e.  R )
)  ->  ( (
( 1st `  F
) `  x )
( 2nd `  (idfunc `  Y
) ) ( ( 1st `  F ) `
 y ) )  =  (  _I  |`  (
( ( 1st `  F
) `  x )
( Hom  `  Y ) ( ( 1st `  F
) `  y )
) ) )
110101, 105, 1093eqtr3d 2478 . . . . . . . 8  |-  ( ( ( ph  /\  F  e.  ( X I Y ) )  /\  (
x  e.  R  /\  y  e.  R )
)  ->  ( (
x ( 2nd `  F
) y )  o.  ( ( ( 1st `  F ) `  x
) ( 2nd `  (
( X (Inv `  C ) Y ) `
 F ) ) ( ( 1st `  F
) `  y )
) )  =  (  _I  |`  ( (
( 1st `  F
) `  x )
( Hom  `  Y ) ( ( 1st `  F
) `  y )
) ) )
11144, 84, 89, 110fcof1od 6207 . . . . . . 7  |-  ( ( ( ph  /\  F  e.  ( X I Y ) )  /\  (
x  e.  R  /\  y  e.  R )
)  ->  ( x
( 2nd `  F
) y ) : ( x ( Hom  `  X ) y ) -1-1-onto-> ( ( ( 1st `  F
) `  x )
( Hom  `  Y ) ( ( 1st `  F
) `  y )
) )
112111ralrimivva 2853 . . . . . 6  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  A. x  e.  R  A. y  e.  R  ( x
( 2nd `  F
) y ) : ( x ( Hom  `  X ) y ) -1-1-onto-> ( ( ( 1st `  F
) `  x )
( Hom  `  Y ) ( ( 1st `  F
) `  y )
) )
11338, 39, 40isffth2 15772 . . . . . 6  |-  ( ( 1st `  F ) ( ( X Full  Y
)  i^i  ( X Faith  Y ) ) ( 2nd `  F )  <->  ( ( 1st `  F ) ( X  Func  Y )
( 2nd `  F
)  /\  A. x  e.  R  A. y  e.  R  ( x
( 2nd `  F
) y ) : ( x ( Hom  `  X ) y ) -1-1-onto-> ( ( ( 1st `  F
) `  x )
( Hom  `  Y ) ( ( 1st `  F
) `  y )
) ) )
11437, 112, 113sylanbrc 668 . . . . 5  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  ( 1st `  F ) ( ( X Full  Y )  i^i  ( X Faith  Y ) ) ( 2nd `  F
) )
115 df-br 4427 . . . . 5  |-  ( ( 1st `  F ) ( ( X Full  Y
)  i^i  ( X Faith  Y ) ) ( 2nd `  F )  <->  <. ( 1st `  F ) ,  ( 2nd `  F )
>.  e.  ( ( X Full 
Y )  i^i  ( X Faith  Y ) ) )
116114, 115sylib 199 . . . 4  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  <. ( 1st `  F ) ,  ( 2nd `  F )
>.  e.  ( ( X Full 
Y )  i^i  ( X Faith  Y ) ) )
11735, 116eqeltrd 2517 . . 3  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  F  e.  ( ( X Full  Y
)  i^i  ( X Faith  Y ) ) )
11838, 45, 37funcf1 15722 . . . 4  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  ( 1st `  F ) : R --> S )
11945, 38, 52funcf1 15722 . . . 4  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  ( 1st `  ( ( X (Inv
`  C ) Y ) `  F ) ) : S --> R )
12064fveq2d 5885 . . . . 5  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  ( 1st `  ( ( ( X (Inv `  C ) Y ) `  F
)  o.func 
F ) )  =  ( 1st `  (idfunc `  X
) ) )
12138, 33, 50cofu1st 15739 . . . . 5  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  ( 1st `  ( ( ( X (Inv `  C ) Y ) `  F
)  o.func 
F ) )  =  ( ( 1st `  (
( X (Inv `  C ) Y ) `
 F ) )  o.  ( 1st `  F
) ) )
12274adantr 466 . . . . . 6  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  X  e.  Cat )
12361, 38, 122idfu1st 15735 . . . . 5  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  ( 1st `  (idfunc `  X ) )  =  (  _I  |`  R ) )
124120, 121, 1233eqtr3d 2478 . . . 4  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  ( ( 1st `  ( ( X (Inv `  C ) Y ) `  F
) )  o.  ( 1st `  F ) )  =  (  _I  |`  R ) )
12598fveq2d 5885 . . . . 5  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  ( 1st `  ( F  o.func  ( ( X (Inv `  C ) Y ) `  F
) ) )  =  ( 1st `  (idfunc `  Y
) ) )
12645, 50, 33cofu1st 15739 . . . . 5  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  ( 1st `  ( F  o.func  ( ( X (Inv `  C ) Y ) `  F
) ) )  =  ( ( 1st `  F
)  o.  ( 1st `  ( ( X (Inv
`  C ) Y ) `  F ) ) ) )
12773, 9sseldd 3471 . . . . . . 7  |-  ( ph  ->  Y  e.  Cat )
128127adantr 466 . . . . . 6  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  Y  e.  Cat )
12995, 45, 128idfu1st 15735 . . . . 5  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  ( 1st `  (idfunc `  Y ) )  =  (  _I  |`  S ) )
130125, 126, 1293eqtr3d 2478 . . . 4  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  ( ( 1st `  F )  o.  ( 1st `  (
( X (Inv `  C ) Y ) `
 F ) ) )  =  (  _I  |`  S ) )
131118, 119, 124, 130fcof1od 6207 . . 3  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  ( 1st `  F ) : R -1-1-onto-> S
)
132117, 131jca 534 . 2  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  ( F  e.  ( ( X Full  Y
)  i^i  ( X Faith  Y ) )  /\  ( 1st `  F ) : R -1-1-onto-> S ) )
1337adantr 466 . . 3  |-  ( (
ph  /\  ( F  e.  ( ( X Full  Y
)  i^i  ( X Faith  Y ) )  /\  ( 1st `  F ) : R -1-1-onto-> S ) )  ->  C  e.  Cat )
1348adantr 466 . . 3  |-  ( (
ph  /\  ( F  e.  ( ( X Full  Y
)  i^i  ( X Faith  Y ) )  /\  ( 1st `  F ) : R -1-1-onto-> S ) )  ->  X  e.  B )
1359adantr 466 . . 3  |-  ( (
ph  /\  ( F  e.  ( ( X Full  Y
)  i^i  ( X Faith  Y ) )  /\  ( 1st `  F ) : R -1-1-onto-> S ) )  ->  Y  e.  B )
136 inss1 3688 . . . . . . 7  |-  ( ( X Full  Y )  i^i  ( X Faith  Y ) )  C_  ( X Full  Y )
137 fullfunc 15762 . . . . . . 7  |-  ( X Full 
Y )  C_  ( X  Func  Y )
138136, 137sstri 3479 . . . . . 6  |-  ( ( X Full  Y )  i^i  ( X Faith  Y ) )  C_  ( X  Func  Y )
139 simprl 762 . . . . . 6  |-  ( (
ph  /\  ( F  e.  ( ( X Full  Y
)  i^i  ( X Faith  Y ) )  /\  ( 1st `  F ) : R -1-1-onto-> S ) )  ->  F  e.  ( ( X Full  Y )  i^i  ( X Faith  Y ) ) )
140138, 139sseldi 3468 . . . . 5  |-  ( (
ph  /\  ( F  e.  ( ( X Full  Y
)  i^i  ( X Faith  Y ) )  /\  ( 1st `  F ) : R -1-1-onto-> S ) )  ->  F  e.  ( X  Func  Y ) )
1411, 140, 34sylancr 667 . . . 4  |-  ( (
ph  /\  ( F  e.  ( ( X Full  Y
)  i^i  ( X Faith  Y ) )  /\  ( 1st `  F ) : R -1-1-onto-> S ) )  ->  F  =  <. ( 1st `  F ) ,  ( 2nd `  F )
>. )
1424adantr 466 . . . . 5  |-  ( (
ph  /\  ( F  e.  ( ( X Full  Y
)  i^i  ( X Faith  Y ) )  /\  ( 1st `  F ) : R -1-1-onto-> S ) )  ->  U  e.  V )
143 eqid 2429 . . . . 5  |-  ( x  e.  S ,  y  e.  S  |->  `' ( ( `' ( 1st `  F ) `  x
) ( 2nd `  F
) ( `' ( 1st `  F ) `
 y ) ) )  =  ( x  e.  S ,  y  e.  S  |->  `' ( ( `' ( 1st `  F ) `  x
) ( 2nd `  F
) ( `' ( 1st `  F ) `
 y ) ) )
144141, 139eqeltrrd 2518 . . . . . 6  |-  ( (
ph  /\  ( F  e.  ( ( X Full  Y
)  i^i  ( X Faith  Y ) )  /\  ( 1st `  F ) : R -1-1-onto-> S ) )  ->  <. ( 1st `  F
) ,  ( 2nd `  F ) >.  e.  ( ( X Full  Y )  i^i  ( X Faith  Y
) ) )
145144, 115sylibr 215 . . . . 5  |-  ( (
ph  /\  ( F  e.  ( ( X Full  Y
)  i^i  ( X Faith  Y ) )  /\  ( 1st `  F ) : R -1-1-onto-> S ) )  -> 
( 1st `  F
) ( ( X Full 
Y )  i^i  ( X Faith  Y ) ) ( 2nd `  F ) )
146 simprr 764 . . . . 5  |-  ( (
ph  /\  ( F  e.  ( ( X Full  Y
)  i^i  ( X Faith  Y ) )  /\  ( 1st `  F ) : R -1-1-onto-> S ) )  -> 
( 1st `  F
) : R -1-1-onto-> S )
1475, 2, 38, 45, 142, 134, 135, 3, 143, 145, 146catcisolem 15952 . . . 4  |-  ( (
ph  /\  ( F  e.  ( ( X Full  Y
)  i^i  ( X Faith  Y ) )  /\  ( 1st `  F ) : R -1-1-onto-> S ) )  ->  <. ( 1st `  F
) ,  ( 2nd `  F ) >. ( X (Inv `  C ) Y ) <. `' ( 1st `  F ) ,  ( x  e.  S ,  y  e.  S  |->  `' ( ( `' ( 1st `  F
) `  x )
( 2nd `  F
) ( `' ( 1st `  F ) `
 y ) ) ) >. )
148141, 147eqbrtrd 4446 . . 3  |-  ( (
ph  /\  ( F  e.  ( ( X Full  Y
)  i^i  ( X Faith  Y ) )  /\  ( 1st `  F ) : R -1-1-onto-> S ) )  ->  F ( X (Inv
`  C ) Y ) <. `' ( 1st `  F ) ,  ( x  e.  S , 
y  e.  S  |->  `' ( ( `' ( 1st `  F ) `
 x ) ( 2nd `  F ) ( `' ( 1st `  F ) `  y
) ) ) >.
)
1492, 3, 133, 134, 135, 10, 148inviso1 15622 . 2  |-  ( (
ph  /\  ( F  e.  ( ( X Full  Y
)  i^i  ( X Faith  Y ) )  /\  ( 1st `  F ) : R -1-1-onto-> S ) )  ->  F  e.  ( X I Y ) )
150132, 149impbida 840 1  |-  ( ph  ->  ( F  e.  ( X I Y )  <-> 
( F  e.  ( ( X Full  Y )  i^i  ( X Faith  Y
) )  /\  ( 1st `  F ) : R -1-1-onto-> S ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1870   A.wral 2782    i^i cin 3441    C_ wss 3442   <.cop 4008   class class class wbr 4426    _I cid 4764   `'ccnv 4853   dom cdm 4854    |` cres 4856    o. ccom 4858   Rel wrel 4859   Fun wfun 5595   -->wf 5597   -1-1-onto->wf1o 5600   ` cfv 5601  (class class class)co 6305    |-> cmpt2 6307   1stc1st 6805   2ndc2nd 6806   Basecbs 15084   Hom chom 15163  compcco 15164   Catccat 15521   Idccid 15522  Sectcsect 15600  Invcinv 15601    Iso ciso 15602    Func cfunc 15710  idfunccidfu 15711    o.func ccofu 15712   Full cful 15758   Faith cfth 15759  CatCatccatc 15940
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-oadd 7194  df-er 7371  df-map 7482  df-ixp 7531  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-10 10676  df-n0 10870  df-z 10938  df-dec 11052  df-uz 11160  df-fz 11783  df-struct 15086  df-ndx 15087  df-slot 15088  df-base 15089  df-hom 15176  df-cco 15177  df-cat 15525  df-cid 15526  df-sect 15603  df-inv 15604  df-iso 15605  df-func 15714  df-idfu 15715  df-cofu 15716  df-full 15760  df-fth 15761  df-catc 15941
This theorem is referenced by:  yoniso  16121
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