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Theorem catciso 14252
Description: A functor is an isomorphism of categories if and only if it is full and faithful, and is a bijection on the objects. (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 14049 . . . . 5  |-  Rel  ( X  Func  Y )
2 catciso.b . . . . . . . . . . . . . 14  |-  B  =  ( Base `  C
)
3 eqid 2435 . . . . . . . . . . . . . 14  |-  (Inv `  C )  =  (Inv
`  C )
4 catciso.u . . . . . . . . . . . . . . 15  |-  ( ph  ->  U  e.  V )
5 catciso.c . . . . . . . . . . . . . . . 16  |-  C  =  (CatCat `  U )
65catccat 14249 . . . . . . . . . . . . . . 15  |-  ( U  e.  V  ->  C  e.  Cat )
74, 6syl 16 . . . . . . . . . . . . . 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 13980 . . . . . . . . . . . . 13  |-  ( ph  ->  ( X I Y )  =  dom  ( X (Inv `  C ) Y ) )
1211eleq2d 2502 . . . . . . . . . . . 12  |-  ( ph  ->  ( F  e.  ( X I Y )  <-> 
F  e.  dom  ( X (Inv `  C ) Y ) ) )
1312biimpa 471 . . . . . . . . . . 11  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  F  e.  dom  ( X (Inv `  C ) Y ) )
147adantr 452 . . . . . . . . . . . . 13  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  C  e.  Cat )
158adantr 452 . . . . . . . . . . . . 13  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  X  e.  B )
169adantr 452 . . . . . . . . . . . . 13  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  Y  e.  B )
172, 3, 14, 15, 16invfun 13979 . . . . . . . . . . . 12  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  Fun  ( X (Inv `  C ) Y ) )
18 funfvbrb 5835 . . . . . . . . . . . 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 16 . . . . . . . . . . 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 202 . . . . . . . . . 10  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  F ( X (Inv `  C ) Y ) ( ( X (Inv `  C
) Y ) `  F ) )
21 eqid 2435 . . . . . . . . . . 11  |-  (Sect `  C )  =  (Sect `  C )
222, 3, 14, 15, 16, 21isinv 13975 . . . . . . . . . 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 202 . . . . . . . . 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 446 . . . . . . . 8  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  F ( X (Sect `  C ) Y ) ( ( X (Inv `  C
) Y ) `  F ) )
25 eqid 2435 . . . . . . . . 9  |-  (  Hom  `  C )  =  (  Hom  `  C )
26 eqid 2435 . . . . . . . . 9  |-  (comp `  C )  =  (comp `  C )
27 eqid 2435 . . . . . . . . 9  |-  ( Id
`  C )  =  ( Id `  C
)
282, 25, 26, 27, 21, 14, 15, 16issect 13969 . . . . . . . 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 202 . . . . . . 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 969 . . . . . 6  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  F  e.  ( X (  Hom  `  C
) Y ) )
315, 2, 4, 25, 8, 9catchom 14244 . . . . . . 7  |-  ( ph  ->  ( X (  Hom  `  C ) Y )  =  ( X  Func  Y ) )
3231adantr 452 . . . . . 6  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  ( X
(  Hom  `  C ) Y )  =  ( X  Func  Y )
)
3330, 32eleqtrd 2511 . . . . 5  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  F  e.  ( X  Func  Y ) )
34 1st2nd 6385 . . . . 5  |-  ( ( Rel  ( X  Func  Y )  /\  F  e.  ( X  Func  Y
) )  ->  F  =  <. ( 1st `  F
) ,  ( 2nd `  F ) >. )
351, 33, 34sylancr 645 . . . 4  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  F  =  <. ( 1st `  F
) ,  ( 2nd `  F ) >. )
36 1st2ndbr 6388 . . . . . . 7  |-  ( ( Rel  ( X  Func  Y )  /\  F  e.  ( X  Func  Y
) )  ->  ( 1st `  F ) ( X  Func  Y )
( 2nd `  F
) )
371, 33, 36sylancr 645 . . . . . 6  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  ( 1st `  F ) ( X 
Func  Y ) ( 2nd `  F ) )
38 catciso.r . . . . . . . . . 10  |-  R  =  ( Base `  X
)
39 eqid 2435 . . . . . . . . . 10  |-  (  Hom  `  X )  =  (  Hom  `  X )
40 eqid 2435 . . . . . . . . . 10  |-  (  Hom  `  Y )  =  (  Hom  `  Y )
4137adantr 452 . . . . . . . . . 10  |-  ( ( ( ph  /\  F  e.  ( X I Y ) )  /\  (
x  e.  R  /\  y  e.  R )
)  ->  ( 1st `  F ) ( X 
Func  Y ) ( 2nd `  F ) )
42 simprl 733 . . . . . . . . . 10  |-  ( ( ( ph  /\  F  e.  ( X I Y ) )  /\  (
x  e.  R  /\  y  e.  R )
)  ->  x  e.  R )
43 simprr 734 . . . . . . . . . 10  |-  ( ( ( ph  /\  F  e.  ( X I Y ) )  /\  (
x  e.  R  /\  y  e.  R )
)  ->  y  e.  R )
4438, 39, 40, 41, 42, 43funcf2 14055 . . . . . . . . 9  |-  ( ( ( 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 . . . . . . . . . . 11  |-  S  =  ( Base `  Y
)
46 relfunc 14049 . . . . . . . . . . . . 13  |-  Rel  ( Y  Func  X )
4729simp2d 970 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  ( ( X (Inv `  C ) Y ) `  F
)  e.  ( Y (  Hom  `  C
) X ) )
485, 2, 4, 25, 9, 8catchom 14244 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( Y (  Hom  `  C ) X )  =  ( Y  Func  X ) )
4948adantr 452 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  ( Y
(  Hom  `  C ) X )  =  ( Y  Func  X )
)
5047, 49eleqtrd 2511 . . . . . . . . . . . . 13  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  ( ( X (Inv `  C ) Y ) `  F
)  e.  ( Y 
Func  X ) )
51 1st2ndbr 6388 . . . . . . . . . . . . 13  |-  ( ( 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 645 . . . . . . . . . . . 12  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  ( 1st `  ( ( X (Inv
`  C ) Y ) `  F ) ) ( Y  Func  X ) ( 2nd `  (
( X (Inv `  C ) Y ) `
 F ) ) )
5352adantr 452 . . . . . . . . . . 11  |-  ( ( ( 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 14053 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  F  e.  ( X I Y ) )  /\  (
x  e.  R  /\  y  e.  R )
)  ->  ( 1st `  F ) : R --> S )
5554, 42ffvelrnd 5863 . . . . . . . . . . 11  |-  ( ( ( ph  /\  F  e.  ( X I Y ) )  /\  (
x  e.  R  /\  y  e.  R )
)  ->  ( ( 1st `  F ) `  x )  e.  S
)
5654, 43ffvelrnd 5863 . . . . . . . . . . 11  |-  ( ( ( ph  /\  F  e.  ( X I Y ) )  /\  (
x  e.  R  /\  y  e.  R )
)  ->  ( ( 1st `  F ) `  y )  e.  S
)
5745, 40, 39, 53, 55, 56funcf2 14055 . . . . . . . . . 10  |-  ( ( ( 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 ) ) ) )
58 eqidd 2436 . . . . . . . . . . 11  |-  ( ( ( ph  /\  F  e.  ( X I Y ) )  /\  (
x  e.  R  /\  y  e.  R )
)  ->  ( (
( 1st `  F
) `  x )
(  Hom  `  Y ) ( ( 1st `  F
) `  y )
)  =  ( ( ( 1st `  F
) `  x )
(  Hom  `  Y ) ( ( 1st `  F
) `  y )
) )
5929simp3d 971 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  ( (
( X (Inv `  C ) Y ) `
 F ) (
<. X ,  Y >. (comp `  C ) X ) F )  =  ( ( Id `  C
) `  X )
)
604adantr 452 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  U  e.  V )
615, 2, 60, 26, 15, 16, 15, 33, 50catcco 14246 . . . . . . . . . . . . . . . . 17  |-  ( (
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 ) )
62 eqid 2435 . . . . . . . . . . . . . . . . . . 19  |-  (idfunc `  X
)  =  (idfunc `  X
)
635, 2, 27, 62, 4, 8catcid 14248 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  ( ( Id `  C ) `  X
)  =  (idfunc `  X
) )
6463adantr 452 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  ( ( Id `  C ) `  X )  =  (idfunc `  X ) )
6559, 61, 643eqtr3d 2475 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  ( (
( X (Inv `  C ) Y ) `
 F )  o.func  F
)  =  (idfunc `  X
) )
6665adantr 452 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  F  e.  ( X I Y ) )  /\  (
x  e.  R  /\  y  e.  R )
)  ->  ( (
( X (Inv `  C ) Y ) `
 F )  o.func  F
)  =  (idfunc `  X
) )
6766fveq2d 5724 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  F  e.  ( X I Y ) )  /\  (
x  e.  R  /\  y  e.  R )
)  ->  ( 1st `  ( ( ( X (Inv `  C ) Y ) `  F
)  o.func 
F ) )  =  ( 1st `  (idfunc `  X
) ) )
6867fveq1d 5722 . . . . . . . . . . . . 13  |-  ( ( ( 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 ) )
6933adantr 452 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  F  e.  ( X I Y ) )  /\  (
x  e.  R  /\  y  e.  R )
)  ->  F  e.  ( X  Func  Y ) )
7050adantr 452 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  F  e.  ( X I Y ) )  /\  (
x  e.  R  /\  y  e.  R )
)  ->  ( ( X (Inv `  C ) Y ) `  F
)  e.  ( Y 
Func  X ) )
7138, 69, 70, 42cofu1 14071 . . . . . . . . . . . . 13  |-  ( ( ( 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 ) ) )
725, 2, 4catcbas 14242 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  B  =  ( U  i^i  Cat ) )
73 inss2 3554 . . . . . . . . . . . . . . . . 17  |-  ( U  i^i  Cat )  C_  Cat
7472, 73syl6eqss 3390 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  B  C_  Cat )
7574, 8sseldd 3341 . . . . . . . . . . . . . . 15  |-  ( ph  ->  X  e.  Cat )
7675ad2antrr 707 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  F  e.  ( X I Y ) )  /\  (
x  e.  R  /\  y  e.  R )
)  ->  X  e.  Cat )
7762, 38, 76, 42idfu1 14067 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  F  e.  ( X I Y ) )  /\  (
x  e.  R  /\  y  e.  R )
)  ->  ( ( 1st `  (idfunc `  X ) ) `  x )  =  x )
7868, 71, 773eqtr3d 2475 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  F  e.  ( X I Y ) )  /\  (
x  e.  R  /\  y  e.  R )
)  ->  ( ( 1st `  ( ( X (Inv `  C ) Y ) `  F
) ) `  (
( 1st `  F
) `  x )
)  =  x )
7967fveq1d 5722 . . . . . . . . . . . . 13  |-  ( ( ( 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 ) )
8038, 69, 70, 43cofu1 14071 . . . . . . . . . . . . 13  |-  ( ( ( 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 ) ) )
8162, 38, 76, 43idfu1 14067 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  F  e.  ( X I Y ) )  /\  (
x  e.  R  /\  y  e.  R )
)  ->  ( ( 1st `  (idfunc `  X ) ) `  y )  =  y )
8279, 80, 813eqtr3d 2475 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  F  e.  ( X I Y ) )  /\  (
x  e.  R  /\  y  e.  R )
)  ->  ( ( 1st `  ( ( X (Inv `  C ) Y ) `  F
) ) `  (
( 1st `  F
) `  y )
)  =  y )
8378, 82oveq12d 6091 . . . . . . . . . . 11  |-  ( ( ( 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 ) )
8458, 83feq23d 5580 . . . . . . . . . 10  |-  ( ( ( 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 ) ) )
8557, 84mpbid 202 . . . . . . . . 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 )
) --> ( x (  Hom  `  X )
y ) )
8623simprd 450 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  ( ( X (Inv `  C ) Y ) `  F
) ( Y (Sect `  C ) X ) F )
872, 25, 26, 27, 21, 14, 16, 15issect 13969 . . . . . . . . . . . . . . . 16  |-  ( (
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 ) ) ) )
8886, 87mpbid 202 . . . . . . . . . . . . . . 15  |-  ( (
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 ) ) )
8988simp3d 971 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  ( F
( <. Y ,  X >. (comp `  C ) Y ) ( ( X (Inv `  C
) Y ) `  F ) )  =  ( ( Id `  C ) `  Y
) )
905, 2, 60, 26, 16, 15, 16, 50, 33catcco 14246 . . . . . . . . . . . . . 14  |-  ( (
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
) ) )
91 eqid 2435 . . . . . . . . . . . . . . . 16  |-  (idfunc `  Y
)  =  (idfunc `  Y
)
925, 2, 27, 91, 4, 9catcid 14248 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( ( Id `  C ) `  Y
)  =  (idfunc `  Y
) )
9392adantr 452 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  ( ( Id `  C ) `  Y )  =  (idfunc `  Y ) )
9489, 90, 933eqtr3d 2475 . . . . . . . . . . . . 13  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  ( F  o.func  ( ( X (Inv
`  C ) Y ) `  F ) )  =  (idfunc `  Y
) )
9594adantr 452 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  F  e.  ( X I Y ) )  /\  (
x  e.  R  /\  y  e.  R )
)  ->  ( F  o.func  ( ( X (Inv
`  C ) Y ) `  F ) )  =  (idfunc `  Y
) )
9695fveq2d 5724 . . . . . . . . . . 11  |-  ( ( ( ph  /\  F  e.  ( X I Y ) )  /\  (
x  e.  R  /\  y  e.  R )
)  ->  ( 2nd `  ( F  o.func  ( ( X (Inv `  C ) Y ) `  F
) ) )  =  ( 2nd `  (idfunc `  Y
) ) )
9796oveqd 6090 . . . . . . . . . 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 `  F ) `
 x ) ( 2nd `  (idfunc `  Y
) ) ( ( 1st `  F ) `
 y ) ) )
9845, 70, 69, 55, 56cofu2nd 14072 . . . . . . . . . . 11  |-  ( ( ( 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 )
) ) )
9978, 82oveq12d 6091 . . . . . . . . . . . 12  |-  ( ( ( 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 ) )
10099coeq1d 5026 . . . . . . . . . . 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 )
) )  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 )
) ) )
10198, 100eqtrd 2467 . . . . . . . . . 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 ) )  =  ( ( x ( 2nd `  F
) y )  o.  ( ( ( 1st `  F ) `  x
) ( 2nd `  (
( X (Inv `  C ) Y ) `
 F ) ) ( ( 1st `  F
) `  y )
) ) )
10274ad2antrr 707 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  F  e.  ( X I Y ) )  /\  (
x  e.  R  /\  y  e.  R )
)  ->  B  C_  Cat )
1039ad2antrr 707 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  F  e.  ( X I Y ) )  /\  (
x  e.  R  /\  y  e.  R )
)  ->  Y  e.  B )
104102, 103sseldd 3341 . . . . . . . . . . 11  |-  ( ( ( ph  /\  F  e.  ( X I Y ) )  /\  (
x  e.  R  /\  y  e.  R )
)  ->  Y  e.  Cat )
10591, 45, 104, 40, 55, 56idfu2nd 14064 . . . . . . . . . 10  |-  ( ( ( 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 )
) ) )
10697, 101, 1053eqtr3d 2475 . . . . . . . . 9  |-  ( ( ( 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 )
) ) )
10766fveq2d 5724 . . . . . . . . . . 11  |-  ( ( ( ph  /\  F  e.  ( X I Y ) )  /\  (
x  e.  R  /\  y  e.  R )
)  ->  ( 2nd `  ( ( ( X (Inv `  C ) Y ) `  F
)  o.func 
F ) )  =  ( 2nd `  (idfunc `  X
) ) )
108107oveqd 6090 . . . . . . . . . 10  |-  ( ( ( 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 ) )
10938, 69, 70, 42, 43cofu2nd 14072 . . . . . . . . . 10  |-  ( ( ( 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 ) ) )
11062, 38, 76, 39, 42, 43idfu2nd 14064 . . . . . . . . . 10  |-  ( ( ( ph  /\  F  e.  ( X I Y ) )  /\  (
x  e.  R  /\  y  e.  R )
)  ->  ( x
( 2nd `  (idfunc `  X
) ) y )  =  (  _I  |`  (
x (  Hom  `  X
) y ) ) )
111108, 109, 1103eqtr3d 2475 . . . . . . . . 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 )
)  o.  ( x ( 2nd `  F
) y ) )  =  (  _I  |`  (
x (  Hom  `  X
) y ) ) )
112 fcof1o 6018 . . . . . . . . 9  |-  ( ( ( ( x ( 2nd `  F ) y ) : ( x (  Hom  `  X
) y ) --> ( ( ( 1st `  F
) `  x )
(  Hom  `  Y ) ( ( 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 ) )  /\  ( ( ( 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 )
) )  /\  (
( ( ( 1st `  F ) `  x
) ( 2nd `  (
( X (Inv `  C ) Y ) `
 F ) ) ( ( 1st `  F
) `  y )
)  o.  ( x ( 2nd `  F
) y ) )  =  (  _I  |`  (
x (  Hom  `  X
) y ) ) ) )  ->  (
( x ( 2nd `  F ) y ) : ( x (  Hom  `  X )
y ) -1-1-onto-> ( ( ( 1st `  F ) `  x
) (  Hom  `  Y
) ( ( 1st `  F ) `  y
) )  /\  `' ( x ( 2nd `  F ) y )  =  ( ( ( 1st `  F ) `
 x ) ( 2nd `  ( ( X (Inv `  C
) Y ) `  F ) ) ( ( 1st `  F
) `  y )
) ) )
11344, 85, 106, 111, 112syl22anc 1185 . . . . . . . 8  |-  ( ( ( 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 )
)  /\  `' (
x ( 2nd `  F
) y )  =  ( ( ( 1st `  F ) `  x
) ( 2nd `  (
( X (Inv `  C ) Y ) `
 F ) ) ( ( 1st `  F
) `  y )
) ) )
114113simpld 446 . . . . . . 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 )
) )
115114ralrimivva 2790 . . . . . 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 )
) )
11638, 39, 40isffth2 14103 . . . . . 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 )
) ) )
11737, 115, 116sylanbrc 646 . . . . 5  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  ( 1st `  F ) ( ( X Full  Y )  i^i  ( X Faith  Y ) ) ( 2nd `  F
) )
118 df-br 4205 . . . . 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 ) ) )
119117, 118sylib 189 . . . 4  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  <. ( 1st `  F ) ,  ( 2nd `  F )
>.  e.  ( ( X Full 
Y )  i^i  ( X Faith  Y ) ) )
12035, 119eqeltrd 2509 . . 3  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  F  e.  ( ( X Full  Y
)  i^i  ( X Faith  Y ) ) )
12138, 45, 37funcf1 14053 . . . . 5  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  ( 1st `  F ) : R --> S )
12245, 38, 52funcf1 14053 . . . . 5  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  ( 1st `  ( ( X (Inv
`  C ) Y ) `  F ) ) : S --> R )
12394fveq2d 5724 . . . . . 6  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  ( 1st `  ( F  o.func  ( ( X (Inv `  C ) Y ) `  F
) ) )  =  ( 1st `  (idfunc `  Y
) ) )
12445, 50, 33cofu1st 14070 . . . . . 6  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  ( 1st `  ( F  o.func  ( ( X (Inv `  C ) Y ) `  F
) ) )  =  ( ( 1st `  F
)  o.  ( 1st `  ( ( X (Inv
`  C ) Y ) `  F ) ) ) )
12574, 9sseldd 3341 . . . . . . . 8  |-  ( ph  ->  Y  e.  Cat )
126125adantr 452 . . . . . . 7  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  Y  e.  Cat )
12791, 45, 126idfu1st 14066 . . . . . 6  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  ( 1st `  (idfunc `  Y ) )  =  (  _I  |`  S ) )
128123, 124, 1273eqtr3d 2475 . . . . 5  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  ( ( 1st `  F )  o.  ( 1st `  (
( X (Inv `  C ) Y ) `
 F ) ) )  =  (  _I  |`  S ) )
12965fveq2d 5724 . . . . . 6  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  ( 1st `  ( ( ( X (Inv `  C ) Y ) `  F
)  o.func 
F ) )  =  ( 1st `  (idfunc `  X
) ) )
13038, 33, 50cofu1st 14070 . . . . . 6  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  ( 1st `  ( ( ( X (Inv `  C ) Y ) `  F
)  o.func 
F ) )  =  ( ( 1st `  (
( X (Inv `  C ) Y ) `
 F ) )  o.  ( 1st `  F
) ) )
13175adantr 452 . . . . . . 7  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  X  e.  Cat )
13262, 38, 131idfu1st 14066 . . . . . 6  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  ( 1st `  (idfunc `  X ) )  =  (  _I  |`  R ) )
133129, 130, 1323eqtr3d 2475 . . . . 5  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  ( ( 1st `  ( ( X (Inv `  C ) Y ) `  F
) )  o.  ( 1st `  F ) )  =  (  _I  |`  R ) )
134 fcof1o 6018 . . . . 5  |-  ( ( ( ( 1st `  F
) : R --> S  /\  ( 1st `  ( ( X (Inv `  C
) Y ) `  F ) ) : S --> R )  /\  ( ( ( 1st `  F )  o.  ( 1st `  ( ( X (Inv `  C ) Y ) `  F
) ) )  =  (  _I  |`  S )  /\  ( ( 1st `  ( ( X (Inv
`  C ) Y ) `  F ) )  o.  ( 1st `  F ) )  =  (  _I  |`  R ) ) )  ->  (
( 1st `  F
) : R -1-1-onto-> S  /\  `' ( 1st `  F
)  =  ( 1st `  ( ( X (Inv
`  C ) Y ) `  F ) ) ) )
135121, 122, 128, 133, 134syl22anc 1185 . . . 4  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  ( ( 1st `  F ) : R -1-1-onto-> S  /\  `' ( 1st `  F )  =  ( 1st `  (
( X (Inv `  C ) Y ) `
 F ) ) ) )
136135simpld 446 . . 3  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  ( 1st `  F ) : R -1-1-onto-> S
)
137120, 136jca 519 . 2  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  ( F  e.  ( ( X Full  Y
)  i^i  ( X Faith  Y ) )  /\  ( 1st `  F ) : R -1-1-onto-> S ) )
1387adantr 452 . . 3  |-  ( (
ph  /\  ( F  e.  ( ( X Full  Y
)  i^i  ( X Faith  Y ) )  /\  ( 1st `  F ) : R -1-1-onto-> S ) )  ->  C  e.  Cat )
1398adantr 452 . . 3  |-  ( (
ph  /\  ( F  e.  ( ( X Full  Y
)  i^i  ( X Faith  Y ) )  /\  ( 1st `  F ) : R -1-1-onto-> S ) )  ->  X  e.  B )
1409adantr 452 . . 3  |-  ( (
ph  /\  ( F  e.  ( ( X Full  Y
)  i^i  ( X Faith  Y ) )  /\  ( 1st `  F ) : R -1-1-onto-> S ) )  ->  Y  e.  B )
141 inss1 3553 . . . . . . 7  |-  ( ( X Full  Y )  i^i  ( X Faith  Y ) )  C_  ( X Full  Y )
142 fullfunc 14093 . . . . . . 7  |-  ( X Full 
Y )  C_  ( X  Func  Y )
143141, 142sstri 3349 . . . . . 6  |-  ( ( X Full  Y )  i^i  ( X Faith  Y ) )  C_  ( X  Func  Y )
144 simprl 733 . . . . . 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 ) ) )
145143, 144sseldi 3338 . . . . 5  |-  ( (
ph  /\  ( F  e.  ( ( X Full  Y
)  i^i  ( X Faith  Y ) )  /\  ( 1st `  F ) : R -1-1-onto-> S ) )  ->  F  e.  ( X  Func  Y ) )
1461, 145, 34sylancr 645 . . . 4  |-  ( (
ph  /\  ( F  e.  ( ( X Full  Y
)  i^i  ( X Faith  Y ) )  /\  ( 1st `  F ) : R -1-1-onto-> S ) )  ->  F  =  <. ( 1st `  F ) ,  ( 2nd `  F )
>. )
1474adantr 452 . . . . 5  |-  ( (
ph  /\  ( F  e.  ( ( X Full  Y
)  i^i  ( X Faith  Y ) )  /\  ( 1st `  F ) : R -1-1-onto-> S ) )  ->  U  e.  V )
148 eqid 2435 . . . . 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 ) ) )
149146, 144eqeltrrd 2510 . . . . . 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
) ) )
150149, 118sylibr 204 . . . . 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 ) )
151 simprr 734 . . . . 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 )
1525, 2, 38, 45, 147, 139, 140, 3, 148, 150, 151catcisolem 14251 . . . 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 ) ) ) >. )
153146, 152eqbrtrd 4224 . . 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
) ) ) >.
)
1542, 3, 138, 139, 140, 10, 153inviso1 13981 . 2  |-  ( (
ph  /\  ( F  e.  ( ( X Full  Y
)  i^i  ( X Faith  Y ) )  /\  ( 1st `  F ) : R -1-1-onto-> S ) )  ->  F  e.  ( X I Y ) )
155137, 154impbida 806 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 set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   A.wral 2697    i^i cin 3311    C_ wss 3312   <.cop 3809   class class class wbr 4204    _I cid 4485   `'ccnv 4869   dom cdm 4870    |` cres 4872    o. ccom 4874   Rel wrel 4875   Fun wfun 5440   -->wf 5442   -1-1-onto->wf1o 5445   ` cfv 5446  (class class class)co 6073    e. cmpt2 6075   1stc1st 6339   2ndc2nd 6340   Basecbs 13459    Hom chom 13530  compcco 13531   Catccat 13879   Idccid 13880  Sectcsect 13960  Invcinv 13961    Iso ciso 13962    Func cfunc 14041  idfunccidfu 14042    o.func ccofu 14043   Full cful 14089   Faith cfth 14090  CatCatccatc 14239
This theorem is referenced by:  yoniso  14372
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9036  ax-resscn 9037  ax-1cn 9038  ax-icn 9039  ax-addcl 9040  ax-addrcl 9041  ax-mulcl 9042  ax-mulrcl 9043  ax-mulcom 9044  ax-addass 9045  ax-mulass 9046  ax-distr 9047  ax-i2m1 9048  ax-1ne0 9049  ax-1rid 9050  ax-rnegex 9051  ax-rrecex 9052  ax-cnre 9053  ax-pre-lttri 9054  ax-pre-lttrn 9055  ax-pre-ltadd 9056  ax-pre-mulgt0 9057
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-map 7012  df-ixp 7056  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-pnf 9112  df-mnf 9113  df-xr 9114  df-ltxr 9115  df-le 9116  df-sub 9283  df-neg 9284  df-nn 9991  df-2 10048  df-3 10049  df-4 10050  df-5 10051  df-6 10052  df-7 10053  df-8 10054  df-9 10055  df-10 10056  df-n0 10212  df-z 10273  df-dec 10373  df-uz 10479  df-fz 11034  df-struct 13461  df-ndx 13462  df-slot 13463  df-base 13464  df-hom 13543  df-cco 13544  df-cat 13883  df-cid 13884  df-sect 13963  df-inv 13964  df-iso 13965  df-func 14045  df-idfu 14046  df-cofu 14047  df-full 14091  df-fth 14092  df-catc 14240
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