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Theorem catciso 14267
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 14064 . . . . 5  |-  Rel  ( X  Func  Y )
2 catciso.b . . . . . . . . . . . . . 14  |-  B  =  ( Base `  C
)
3 eqid 2438 . . . . . . . . . . . . . 14  |-  (Inv `  C )  =  (Inv
`  C )
4 catciso.u . . . . . . . . . . . . . . 15  |-  ( ph  ->  U  e.  V )
5 catciso.c . . . . . . . . . . . . . . . 16  |-  C  =  (CatCat `  U )
65catccat 14264 . . . . . . . . . . . . . . 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 13995 . . . . . . . . . . . . 13  |-  ( ph  ->  ( X I Y )  =  dom  ( X (Inv `  C ) Y ) )
1211eleq2d 2505 . . . . . . . . . . . 12  |-  ( ph  ->  ( F  e.  ( X I Y )  <-> 
F  e.  dom  ( X (Inv `  C ) Y ) ) )
1312biimpa 472 . . . . . . . . . . 11  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  F  e.  dom  ( X (Inv `  C ) Y ) )
147adantr 453 . . . . . . . . . . . . 13  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  C  e.  Cat )
158adantr 453 . . . . . . . . . . . . 13  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  X  e.  B )
169adantr 453 . . . . . . . . . . . . 13  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  Y  e.  B )
172, 3, 14, 15, 16invfun 13994 . . . . . . . . . . . 12  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  Fun  ( X (Inv `  C ) Y ) )
18 funfvbrb 5846 . . . . . . . . . . . 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 203 . . . . . . . . . 10  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  F ( X (Inv `  C ) Y ) ( ( X (Inv `  C
) Y ) `  F ) )
21 eqid 2438 . . . . . . . . . . 11  |-  (Sect `  C )  =  (Sect `  C )
222, 3, 14, 15, 16, 21isinv 13990 . . . . . . . . . 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 203 . . . . . . . . 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 447 . . . . . . . 8  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  F ( X (Sect `  C ) Y ) ( ( X (Inv `  C
) Y ) `  F ) )
25 eqid 2438 . . . . . . . . 9  |-  (  Hom  `  C )  =  (  Hom  `  C )
26 eqid 2438 . . . . . . . . 9  |-  (comp `  C )  =  (comp `  C )
27 eqid 2438 . . . . . . . . 9  |-  ( Id
`  C )  =  ( Id `  C
)
282, 25, 26, 27, 21, 14, 15, 16issect 13984 . . . . . . . 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 203 . . . . . . 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 970 . . . . . 6  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  F  e.  ( X (  Hom  `  C
) Y ) )
315, 2, 4, 25, 8, 9catchom 14259 . . . . . . 7  |-  ( ph  ->  ( X (  Hom  `  C ) Y )  =  ( X  Func  Y ) )
3231adantr 453 . . . . . 6  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  ( X
(  Hom  `  C ) Y )  =  ( X  Func  Y )
)
3330, 32eleqtrd 2514 . . . . 5  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  F  e.  ( X  Func  Y ) )
34 1st2nd 6396 . . . . 5  |-  ( ( Rel  ( X  Func  Y )  /\  F  e.  ( X  Func  Y
) )  ->  F  =  <. ( 1st `  F
) ,  ( 2nd `  F ) >. )
351, 33, 34sylancr 646 . . . 4  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  F  =  <. ( 1st `  F
) ,  ( 2nd `  F ) >. )
36 1st2ndbr 6399 . . . . . . 7  |-  ( ( Rel  ( X  Func  Y )  /\  F  e.  ( X  Func  Y
) )  ->  ( 1st `  F ) ( X  Func  Y )
( 2nd `  F
) )
371, 33, 36sylancr 646 . . . . . 6  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  ( 1st `  F ) ( X 
Func  Y ) ( 2nd `  F ) )
38 catciso.r . . . . . . . . . 10  |-  R  =  ( Base `  X
)
39 eqid 2438 . . . . . . . . . 10  |-  (  Hom  `  X )  =  (  Hom  `  X )
40 eqid 2438 . . . . . . . . . 10  |-  (  Hom  `  Y )  =  (  Hom  `  Y )
4137adantr 453 . . . . . . . . . 10  |-  ( ( ( ph  /\  F  e.  ( X I Y ) )  /\  (
x  e.  R  /\  y  e.  R )
)  ->  ( 1st `  F ) ( X 
Func  Y ) ( 2nd `  F ) )
42 simprl 734 . . . . . . . . . 10  |-  ( ( ( ph  /\  F  e.  ( X I Y ) )  /\  (
x  e.  R  /\  y  e.  R )
)  ->  x  e.  R )
43 simprr 735 . . . . . . . . . 10  |-  ( ( ( ph  /\  F  e.  ( X I Y ) )  /\  (
x  e.  R  /\  y  e.  R )
)  ->  y  e.  R )
4438, 39, 40, 41, 42, 43funcf2 14070 . . . . . . . . 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 14064 . . . . . . . . . . . . 13  |-  Rel  ( Y  Func  X )
4729simp2d 971 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  ( ( X (Inv `  C ) Y ) `  F
)  e.  ( Y (  Hom  `  C
) X ) )
485, 2, 4, 25, 9, 8catchom 14259 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( Y (  Hom  `  C ) X )  =  ( Y  Func  X ) )
4948adantr 453 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  ( Y
(  Hom  `  C ) X )  =  ( Y  Func  X )
)
5047, 49eleqtrd 2514 . . . . . . . . . . . . 13  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  ( ( X (Inv `  C ) Y ) `  F
)  e.  ( Y 
Func  X ) )
51 1st2ndbr 6399 . . . . . . . . . . . . 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 646 . . . . . . . . . . . 12  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  ( 1st `  ( ( X (Inv
`  C ) Y ) `  F ) ) ( Y  Func  X ) ( 2nd `  (
( X (Inv `  C ) Y ) `
 F ) ) )
5352adantr 453 . . . . . . . . . . 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 14068 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  F  e.  ( X I Y ) )  /\  (
x  e.  R  /\  y  e.  R )
)  ->  ( 1st `  F ) : R --> S )
5554, 42ffvelrnd 5874 . . . . . . . . . . 11  |-  ( ( ( ph  /\  F  e.  ( X I Y ) )  /\  (
x  e.  R  /\  y  e.  R )
)  ->  ( ( 1st `  F ) `  x )  e.  S
)
5654, 43ffvelrnd 5874 . . . . . . . . . . 11  |-  ( ( ( ph  /\  F  e.  ( X I Y ) )  /\  (
x  e.  R  /\  y  e.  R )
)  ->  ( ( 1st `  F ) `  y )  e.  S
)
5745, 40, 39, 53, 55, 56funcf2 14070 . . . . . . . . . 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 2439 . . . . . . . . . . 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 972 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  ( (
( X (Inv `  C ) Y ) `
 F ) (
<. X ,  Y >. (comp `  C ) X ) F )  =  ( ( Id `  C
) `  X )
)
604adantr 453 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  U  e.  V )
615, 2, 60, 26, 15, 16, 15, 33, 50catcco 14261 . . . . . . . . . . . . . . . . 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 2438 . . . . . . . . . . . . . . . . . . 19  |-  (idfunc `  X
)  =  (idfunc `  X
)
635, 2, 27, 62, 4, 8catcid 14263 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  ( ( Id `  C ) `  X
)  =  (idfunc `  X
) )
6463adantr 453 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  ( ( Id `  C ) `  X )  =  (idfunc `  X ) )
6559, 61, 643eqtr3d 2478 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  ( (
( X (Inv `  C ) Y ) `
 F )  o.func  F
)  =  (idfunc `  X
) )
6665adantr 453 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  F  e.  ( X I Y ) )  /\  (
x  e.  R  /\  y  e.  R )
)  ->  ( (
( X (Inv `  C ) Y ) `
 F )  o.func  F
)  =  (idfunc `  X
) )
6766fveq2d 5735 . . . . . . . . . . . . . 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 5733 . . . . . . . . . . . . 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 453 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  F  e.  ( X I Y ) )  /\  (
x  e.  R  /\  y  e.  R )
)  ->  F  e.  ( X  Func  Y ) )
7050adantr 453 . . . . . . . . . . . . . 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 14086 . . . . . . . . . . . . 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 14257 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  B  =  ( U  i^i  Cat ) )
73 inss2 3564 . . . . . . . . . . . . . . . . 17  |-  ( U  i^i  Cat )  C_  Cat
7472, 73syl6eqss 3400 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  B  C_  Cat )
7574, 8sseldd 3351 . . . . . . . . . . . . . . 15  |-  ( ph  ->  X  e.  Cat )
7675ad2antrr 708 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  F  e.  ( X I Y ) )  /\  (
x  e.  R  /\  y  e.  R )
)  ->  X  e.  Cat )
7762, 38, 76, 42idfu1 14082 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  F  e.  ( X I Y ) )  /\  (
x  e.  R  /\  y  e.  R )
)  ->  ( ( 1st `  (idfunc `  X ) ) `  x )  =  x )
7868, 71, 773eqtr3d 2478 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  F  e.  ( X I Y ) )  /\  (
x  e.  R  /\  y  e.  R )
)  ->  ( ( 1st `  ( ( X (Inv `  C ) Y ) `  F
) ) `  (
( 1st `  F
) `  x )
)  =  x )
7967fveq1d 5733 . . . . . . . . . . . . 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 14086 . . . . . . . . . . . . 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 14082 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  F  e.  ( X I Y ) )  /\  (
x  e.  R  /\  y  e.  R )
)  ->  ( ( 1st `  (idfunc `  X ) ) `  y )  =  y )
8279, 80, 813eqtr3d 2478 . . . . . . . . . . . 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 6102 . . . . . . . . . . 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 5591 . . . . . . . . . 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 203 . . . . . . . . 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 451 . . . . . . . . . . . . . . . 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 13984 . . . . . . . . . . . . . . . 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 203 . . . . . . . . . . . . . . 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 972 . . . . . . . . . . . . . 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 14261 . . . . . . . . . . . . . 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 2438 . . . . . . . . . . . . . . . 16  |-  (idfunc `  Y
)  =  (idfunc `  Y
)
925, 2, 27, 91, 4, 9catcid 14263 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( ( Id `  C ) `  Y
)  =  (idfunc `  Y
) )
9392adantr 453 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  ( ( Id `  C ) `  Y )  =  (idfunc `  Y ) )
9489, 90, 933eqtr3d 2478 . . . . . . . . . . . . 13  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  ( F  o.func  ( ( X (Inv
`  C ) Y ) `  F ) )  =  (idfunc `  Y
) )
9594adantr 453 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  F  e.  ( X I Y ) )  /\  (
x  e.  R  /\  y  e.  R )
)  ->  ( F  o.func  ( ( X (Inv
`  C ) Y ) `  F ) )  =  (idfunc `  Y
) )
9695fveq2d 5735 . . . . . . . . . . 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 6101 . . . . . . . . . 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 14087 . . . . . . . . . . 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 6102 . . . . . . . . . . . 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 5037 . . . . . . . . . . 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 2470 . . . . . . . . . 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 708 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  F  e.  ( X I Y ) )  /\  (
x  e.  R  /\  y  e.  R )
)  ->  B  C_  Cat )
1039ad2antrr 708 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  F  e.  ( X I Y ) )  /\  (
x  e.  R  /\  y  e.  R )
)  ->  Y  e.  B )
104102, 103sseldd 3351 . . . . . . . . . . 11  |-  ( ( ( ph  /\  F  e.  ( X I Y ) )  /\  (
x  e.  R  /\  y  e.  R )
)  ->  Y  e.  Cat )
10591, 45, 104, 40, 55, 56idfu2nd 14079 . . . . . . . . . 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 2478 . . . . . . . . 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 5735 . . . . . . . . . . 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 6101 . . . . . . . . . 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 14087 . . . . . . . . . 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 14079 . . . . . . . . . 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 2478 . . . . . . . . 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 6029 . . . . . . . . 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 1186 . . . . . . . 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 447 . . . . . . 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 2800 . . . . . 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 14118 . . . . . 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 647 . . . . 5  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  ( 1st `  F ) ( ( X Full  Y )  i^i  ( X Faith  Y ) ) ( 2nd `  F
) )
118 df-br 4216 . . . . 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 190 . . . 4  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  <. ( 1st `  F ) ,  ( 2nd `  F )
>.  e.  ( ( X Full 
Y )  i^i  ( X Faith  Y ) ) )
12035, 119eqeltrd 2512 . . 3  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  F  e.  ( ( X Full  Y
)  i^i  ( X Faith  Y ) ) )
12138, 45, 37funcf1 14068 . . . . 5  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  ( 1st `  F ) : R --> S )
12245, 38, 52funcf1 14068 . . . . 5  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  ( 1st `  ( ( X (Inv
`  C ) Y ) `  F ) ) : S --> R )
12394fveq2d 5735 . . . . . 6  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  ( 1st `  ( F  o.func  ( ( X (Inv `  C ) Y ) `  F
) ) )  =  ( 1st `  (idfunc `  Y
) ) )
12445, 50, 33cofu1st 14085 . . . . . 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 3351 . . . . . . . 8  |-  ( ph  ->  Y  e.  Cat )
126125adantr 453 . . . . . . 7  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  Y  e.  Cat )
12791, 45, 126idfu1st 14081 . . . . . 6  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  ( 1st `  (idfunc `  Y ) )  =  (  _I  |`  S ) )
128123, 124, 1273eqtr3d 2478 . . . . 5  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  ( ( 1st `  F )  o.  ( 1st `  (
( X (Inv `  C ) Y ) `
 F ) ) )  =  (  _I  |`  S ) )
12965fveq2d 5735 . . . . . 6  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  ( 1st `  ( ( ( X (Inv `  C ) Y ) `  F
)  o.func 
F ) )  =  ( 1st `  (idfunc `  X
) ) )
13038, 33, 50cofu1st 14085 . . . . . 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 453 . . . . . . 7  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  X  e.  Cat )
13262, 38, 131idfu1st 14081 . . . . . 6  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  ( 1st `  (idfunc `  X ) )  =  (  _I  |`  R ) )
133129, 130, 1323eqtr3d 2478 . . . . 5  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  ( ( 1st `  ( ( X (Inv `  C ) Y ) `  F
) )  o.  ( 1st `  F ) )  =  (  _I  |`  R ) )
134 fcof1o 6029 . . . . 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 1186 . . . 4  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  ( ( 1st `  F ) : R -1-1-onto-> S  /\  `' ( 1st `  F )  =  ( 1st `  (
( X (Inv `  C ) Y ) `
 F ) ) ) )
136135simpld 447 . . 3  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  ( 1st `  F ) : R -1-1-onto-> S
)
137120, 136jca 520 . 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 453 . . 3  |-  ( (
ph  /\  ( F  e.  ( ( X Full  Y
)  i^i  ( X Faith  Y ) )  /\  ( 1st `  F ) : R -1-1-onto-> S ) )  ->  C  e.  Cat )
1398adantr 453 . . 3  |-  ( (
ph  /\  ( F  e.  ( ( X Full  Y
)  i^i  ( X Faith  Y ) )  /\  ( 1st `  F ) : R -1-1-onto-> S ) )  ->  X  e.  B )
1409adantr 453 . . 3  |-  ( (
ph  /\  ( F  e.  ( ( X Full  Y
)  i^i  ( X Faith  Y ) )  /\  ( 1st `  F ) : R -1-1-onto-> S ) )  ->  Y  e.  B )
141 inss1 3563 . . . . . . 7  |-  ( ( X Full  Y )  i^i  ( X Faith  Y ) )  C_  ( X Full  Y )
142 fullfunc 14108 . . . . . . 7  |-  ( X Full 
Y )  C_  ( X  Func  Y )
143141, 142sstri 3359 . . . . . 6  |-  ( ( X Full  Y )  i^i  ( X Faith  Y ) )  C_  ( X  Func  Y )
144 simprl 734 . . . . . 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 3348 . . . . 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 646 . . . 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 453 . . . . 5  |-  ( (
ph  /\  ( F  e.  ( ( X Full  Y
)  i^i  ( X Faith  Y ) )  /\  ( 1st `  F ) : R -1-1-onto-> S ) )  ->  U  e.  V )
148 eqid 2438 . . . . 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 2513 . . . . . 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 205 . . . . 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 735 . . . . 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 14266 . . . 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 4235 . . 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 13996 . 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 807 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 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   A.wral 2707    i^i cin 3321    C_ wss 3322   <.cop 3819   class class class wbr 4215    _I cid 4496   `'ccnv 4880   dom cdm 4881    |` cres 4883    o. ccom 4885   Rel wrel 4886   Fun wfun 5451   -->wf 5453   -1-1-onto->wf1o 5456   ` cfv 5457  (class class class)co 6084    e. cmpt2 6086   1stc1st 6350   2ndc2nd 6351   Basecbs 13474    Hom chom 13545  compcco 13546   Catccat 13894   Idccid 13895  Sectcsect 13975  Invcinv 13976    Iso ciso 13977    Func cfunc 14056  idfunccidfu 14057    o.func ccofu 14058   Full cful 14104   Faith cfth 14105  CatCatccatc 14254
This theorem is referenced by:  yoniso  14387
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-riota 6552  df-recs 6636  df-rdg 6671  df-1o 6727  df-oadd 6731  df-er 6908  df-map 7023  df-ixp 7067  df-en 7113  df-dom 7114  df-sdom 7115  df-fin 7116  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-nn 10006  df-2 10063  df-3 10064  df-4 10065  df-5 10066  df-6 10067  df-7 10068  df-8 10069  df-9 10070  df-10 10071  df-n0 10227  df-z 10288  df-dec 10388  df-uz 10494  df-fz 11049  df-struct 13476  df-ndx 13477  df-slot 13478  df-base 13479  df-hom 13558  df-cco 13559  df-cat 13898  df-cid 13899  df-sect 13978  df-inv 13979  df-iso 13980  df-func 14060  df-idfu 14061  df-cofu 14062  df-full 14106  df-fth 14107  df-catc 14255
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