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Theorem catciso 13939
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 13736 . . . . 5  |-  Rel  ( X  Func  Y )
2 catciso.b . . . . . . . . . . . . . 14  |-  B  =  ( Base `  C
)
3 eqid 2283 . . . . . . . . . . . . . 14  |-  (Inv `  C )  =  (Inv
`  C )
4 catciso.u . . . . . . . . . . . . . . 15  |-  ( ph  ->  U  e.  V )
5 catciso.c . . . . . . . . . . . . . . . 16  |-  C  =  (CatCat `  U )
65catccat 13936 . . . . . . . . . . . . . . 15  |-  ( U  e.  V  ->  C  e.  Cat )
74, 6syl 15 . . . . . . . . . . . . . 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 13667 . . . . . . . . . . . . 13  |-  ( ph  ->  ( X I Y )  =  dom  ( X (Inv `  C ) Y ) )
1211eleq2d 2350 . . . . . . . . . . . 12  |-  ( ph  ->  ( F  e.  ( X I Y )  <-> 
F  e.  dom  ( X (Inv `  C ) Y ) ) )
1312biimpa 470 . . . . . . . . . . 11  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  F  e.  dom  ( X (Inv `  C ) Y ) )
147adantr 451 . . . . . . . . . . . . 13  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  C  e.  Cat )
158adantr 451 . . . . . . . . . . . . 13  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  X  e.  B )
169adantr 451 . . . . . . . . . . . . 13  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  Y  e.  B )
172, 3, 14, 15, 16invfun 13666 . . . . . . . . . . . 12  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  Fun  ( X (Inv `  C ) Y ) )
18 funfvbrb 5638 . . . . . . . . . . . 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 15 . . . . . . . . . . 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 201 . . . . . . . . . 10  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  F ( X (Inv `  C ) Y ) ( ( X (Inv `  C
) Y ) `  F ) )
21 eqid 2283 . . . . . . . . . . 11  |-  (Sect `  C )  =  (Sect `  C )
222, 3, 14, 15, 16, 21isinv 13662 . . . . . . . . . 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 201 . . . . . . . . 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 445 . . . . . . . 8  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  F ( X (Sect `  C ) Y ) ( ( X (Inv `  C
) Y ) `  F ) )
25 eqid 2283 . . . . . . . . 9  |-  (  Hom  `  C )  =  (  Hom  `  C )
26 eqid 2283 . . . . . . . . 9  |-  (comp `  C )  =  (comp `  C )
27 eqid 2283 . . . . . . . . 9  |-  ( Id
`  C )  =  ( Id `  C
)
282, 25, 26, 27, 21, 14, 15, 16issect 13656 . . . . . . . 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 201 . . . . . . 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 967 . . . . . 6  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  F  e.  ( X (  Hom  `  C
) Y ) )
315, 2, 4, 25, 8, 9catchom 13931 . . . . . . 7  |-  ( ph  ->  ( X (  Hom  `  C ) Y )  =  ( X  Func  Y ) )
3231adantr 451 . . . . . 6  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  ( X
(  Hom  `  C ) Y )  =  ( X  Func  Y )
)
3330, 32eleqtrd 2359 . . . . 5  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  F  e.  ( X  Func  Y ) )
34 1st2nd 6166 . . . . 5  |-  ( ( Rel  ( X  Func  Y )  /\  F  e.  ( X  Func  Y
) )  ->  F  =  <. ( 1st `  F
) ,  ( 2nd `  F ) >. )
351, 33, 34sylancr 644 . . . 4  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  F  =  <. ( 1st `  F
) ,  ( 2nd `  F ) >. )
36 1st2ndbr 6169 . . . . . . 7  |-  ( ( Rel  ( X  Func  Y )  /\  F  e.  ( X  Func  Y
) )  ->  ( 1st `  F ) ( X  Func  Y )
( 2nd `  F
) )
371, 33, 36sylancr 644 . . . . . 6  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  ( 1st `  F ) ( X 
Func  Y ) ( 2nd `  F ) )
38 catciso.r . . . . . . . . . 10  |-  R  =  ( Base `  X
)
39 eqid 2283 . . . . . . . . . 10  |-  (  Hom  `  X )  =  (  Hom  `  X )
40 eqid 2283 . . . . . . . . . 10  |-  (  Hom  `  Y )  =  (  Hom  `  Y )
4137adantr 451 . . . . . . . . . 10  |-  ( ( ( ph  /\  F  e.  ( X I Y ) )  /\  (
x  e.  R  /\  y  e.  R )
)  ->  ( 1st `  F ) ( X 
Func  Y ) ( 2nd `  F ) )
42 simprl 732 . . . . . . . . . 10  |-  ( ( ( ph  /\  F  e.  ( X I Y ) )  /\  (
x  e.  R  /\  y  e.  R )
)  ->  x  e.  R )
43 simprr 733 . . . . . . . . . 10  |-  ( ( ( ph  /\  F  e.  ( X I Y ) )  /\  (
x  e.  R  /\  y  e.  R )
)  ->  y  e.  R )
4438, 39, 40, 41, 42, 43funcf2 13742 . . . . . . . . 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 13736 . . . . . . . . . . . . 13  |-  Rel  ( Y  Func  X )
4729simp2d 968 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  ( ( X (Inv `  C ) Y ) `  F
)  e.  ( Y (  Hom  `  C
) X ) )
485, 2, 4, 25, 9, 8catchom 13931 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( Y (  Hom  `  C ) X )  =  ( Y  Func  X ) )
4948adantr 451 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  ( Y
(  Hom  `  C ) X )  =  ( Y  Func  X )
)
5047, 49eleqtrd 2359 . . . . . . . . . . . . 13  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  ( ( X (Inv `  C ) Y ) `  F
)  e.  ( Y 
Func  X ) )
51 1st2ndbr 6169 . . . . . . . . . . . . 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 644 . . . . . . . . . . . 12  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  ( 1st `  ( ( X (Inv
`  C ) Y ) `  F ) ) ( Y  Func  X ) ( 2nd `  (
( X (Inv `  C ) Y ) `
 F ) ) )
5352adantr 451 . . . . . . . . . . 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 13740 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  F  e.  ( X I Y ) )  /\  (
x  e.  R  /\  y  e.  R )
)  ->  ( 1st `  F ) : R --> S )
5554, 42ffvelrnd 5666 . . . . . . . . . . 11  |-  ( ( ( ph  /\  F  e.  ( X I Y ) )  /\  (
x  e.  R  /\  y  e.  R )
)  ->  ( ( 1st `  F ) `  x )  e.  S
)
5654, 43ffvelrnd 5666 . . . . . . . . . . 11  |-  ( ( ( ph  /\  F  e.  ( X I Y ) )  /\  (
x  e.  R  /\  y  e.  R )
)  ->  ( ( 1st `  F ) `  y )  e.  S
)
5745, 40, 39, 53, 55, 56funcf2 13742 . . . . . . . . . 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 2284 . . . . . . . . . . 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 969 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  ( (
( X (Inv `  C ) Y ) `
 F ) (
<. X ,  Y >. (comp `  C ) X ) F )  =  ( ( Id `  C
) `  X )
)
604adantr 451 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  U  e.  V )
615, 2, 60, 26, 15, 16, 15, 33, 50catcco 13933 . . . . . . . . . . . . . . . . 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 2283 . . . . . . . . . . . . . . . . . . 19  |-  (idfunc `  X
)  =  (idfunc `  X
)
635, 2, 27, 62, 4, 8catcid 13935 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  ( ( Id `  C ) `  X
)  =  (idfunc `  X
) )
6463adantr 451 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  ( ( Id `  C ) `  X )  =  (idfunc `  X ) )
6559, 61, 643eqtr3d 2323 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  ( (
( X (Inv `  C ) Y ) `
 F )  o.func  F
)  =  (idfunc `  X
) )
6665adantr 451 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  F  e.  ( X I Y ) )  /\  (
x  e.  R  /\  y  e.  R )
)  ->  ( (
( X (Inv `  C ) Y ) `
 F )  o.func  F
)  =  (idfunc `  X
) )
6766fveq2d 5529 . . . . . . . . . . . . . 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 5527 . . . . . . . . . . . . 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 451 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  F  e.  ( X I Y ) )  /\  (
x  e.  R  /\  y  e.  R )
)  ->  F  e.  ( X  Func  Y ) )
7050adantr 451 . . . . . . . . . . . . . 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 13758 . . . . . . . . . . . . 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 13929 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  B  =  ( U  i^i  Cat ) )
73 inss2 3390 . . . . . . . . . . . . . . . . 17  |-  ( U  i^i  Cat )  C_  Cat
7472, 73syl6eqss 3228 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  B  C_  Cat )
7574, 8sseldd 3181 . . . . . . . . . . . . . . 15  |-  ( ph  ->  X  e.  Cat )
7675ad2antrr 706 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  F  e.  ( X I Y ) )  /\  (
x  e.  R  /\  y  e.  R )
)  ->  X  e.  Cat )
7762, 38, 76, 42idfu1 13754 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  F  e.  ( X I Y ) )  /\  (
x  e.  R  /\  y  e.  R )
)  ->  ( ( 1st `  (idfunc `  X ) ) `  x )  =  x )
7868, 71, 773eqtr3d 2323 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  F  e.  ( X I Y ) )  /\  (
x  e.  R  /\  y  e.  R )
)  ->  ( ( 1st `  ( ( X (Inv `  C ) Y ) `  F
) ) `  (
( 1st `  F
) `  x )
)  =  x )
7967fveq1d 5527 . . . . . . . . . . . . 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 13758 . . . . . . . . . . . . 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 13754 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  F  e.  ( X I Y ) )  /\  (
x  e.  R  /\  y  e.  R )
)  ->  ( ( 1st `  (idfunc `  X ) ) `  y )  =  y )
8279, 80, 813eqtr3d 2323 . . . . . . . . . . . 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 5876 . . . . . . . . . . 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 5386 . . . . . . . . . 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 201 . . . . . . . . 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 449 . . . . . . . . . . . . . . . 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 13656 . . . . . . . . . . . . . . . 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 201 . . . . . . . . . . . . . . 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 969 . . . . . . . . . . . . . 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 13933 . . . . . . . . . . . . . 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 2283 . . . . . . . . . . . . . . . 16  |-  (idfunc `  Y
)  =  (idfunc `  Y
)
925, 2, 27, 91, 4, 9catcid 13935 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( ( Id `  C ) `  Y
)  =  (idfunc `  Y
) )
9392adantr 451 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  ( ( Id `  C ) `  Y )  =  (idfunc `  Y ) )
9489, 90, 933eqtr3d 2323 . . . . . . . . . . . . 13  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  ( F  o.func  ( ( X (Inv
`  C ) Y ) `  F ) )  =  (idfunc `  Y
) )
9594adantr 451 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  F  e.  ( X I Y ) )  /\  (
x  e.  R  /\  y  e.  R )
)  ->  ( F  o.func  ( ( X (Inv
`  C ) Y ) `  F ) )  =  (idfunc `  Y
) )
9695fveq2d 5529 . . . . . . . . . . 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 5875 . . . . . . . . . 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 13759 . . . . . . . . . . 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 5876 . . . . . . . . . . . 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 4845 . . . . . . . . . . 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 2315 . . . . . . . . . 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 706 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  F  e.  ( X I Y ) )  /\  (
x  e.  R  /\  y  e.  R )
)  ->  B  C_  Cat )
10316adantr 451 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  F  e.  ( X I Y ) )  /\  (
x  e.  R  /\  y  e.  R )
)  ->  Y  e.  B )
104102, 103sseldd 3181 . . . . . . . . . . 11  |-  ( ( ( ph  /\  F  e.  ( X I Y ) )  /\  (
x  e.  R  /\  y  e.  R )
)  ->  Y  e.  Cat )
10591, 45, 104, 40, 55, 56idfu2nd 13751 . . . . . . . . . 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 2323 . . . . . . . . 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 5529 . . . . . . . . . . 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 5875 . . . . . . . . . 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 13759 . . . . . . . . . 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 13751 . . . . . . . . . 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 2323 . . . . . . . . 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 5803 . . . . . . . . 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 1183 . . . . . . . 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 445 . . . . . . 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 2635 . . . . . 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 13790 . . . . . 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 645 . . . . 5  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  ( 1st `  F ) ( ( X Full  Y )  i^i  ( X Faith  Y ) ) ( 2nd `  F
) )
118 df-br 4024 . . . . 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 188 . . . 4  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  <. ( 1st `  F ) ,  ( 2nd `  F )
>.  e.  ( ( X Full 
Y )  i^i  ( X Faith  Y ) ) )
12035, 119eqeltrd 2357 . . 3  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  F  e.  ( ( X Full  Y
)  i^i  ( X Faith  Y ) ) )
12138, 45, 37funcf1 13740 . . . . 5  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  ( 1st `  F ) : R --> S )
12245, 38, 52funcf1 13740 . . . . 5  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  ( 1st `  ( ( X (Inv
`  C ) Y ) `  F ) ) : S --> R )
12394fveq2d 5529 . . . . . 6  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  ( 1st `  ( F  o.func  ( ( X (Inv `  C ) Y ) `  F
) ) )  =  ( 1st `  (idfunc `  Y
) ) )
12445, 50, 33cofu1st 13757 . . . . . 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 3181 . . . . . . . 8  |-  ( ph  ->  Y  e.  Cat )
126125adantr 451 . . . . . . 7  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  Y  e.  Cat )
12791, 45, 126idfu1st 13753 . . . . . 6  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  ( 1st `  (idfunc `  Y ) )  =  (  _I  |`  S ) )
128123, 124, 1273eqtr3d 2323 . . . . 5  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  ( ( 1st `  F )  o.  ( 1st `  (
( X (Inv `  C ) Y ) `
 F ) ) )  =  (  _I  |`  S ) )
12965fveq2d 5529 . . . . . 6  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  ( 1st `  ( ( ( X (Inv `  C ) Y ) `  F
)  o.func 
F ) )  =  ( 1st `  (idfunc `  X
) ) )
13038, 33, 50cofu1st 13757 . . . . . 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 451 . . . . . . 7  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  X  e.  Cat )
13262, 38, 131idfu1st 13753 . . . . . 6  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  ( 1st `  (idfunc `  X ) )  =  (  _I  |`  R ) )
133129, 130, 1323eqtr3d 2323 . . . . 5  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  ( ( 1st `  ( ( X (Inv `  C ) Y ) `  F
) )  o.  ( 1st `  F ) )  =  (  _I  |`  R ) )
134 fcof1o 5803 . . . . 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 1183 . . . 4  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  ( ( 1st `  F ) : R -1-1-onto-> S  /\  `' ( 1st `  F )  =  ( 1st `  (
( X (Inv `  C ) Y ) `
 F ) ) ) )
136135simpld 445 . . 3  |-  ( (
ph  /\  F  e.  ( X I Y ) )  ->  ( 1st `  F ) : R -1-1-onto-> S
)
137120, 136jca 518 . 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 451 . . 3  |-  ( (
ph  /\  ( F  e.  ( ( X Full  Y
)  i^i  ( X Faith  Y ) )  /\  ( 1st `  F ) : R -1-1-onto-> S ) )  ->  C  e.  Cat )
1398adantr 451 . . 3  |-  ( (
ph  /\  ( F  e.  ( ( X Full  Y
)  i^i  ( X Faith  Y ) )  /\  ( 1st `  F ) : R -1-1-onto-> S ) )  ->  X  e.  B )
1409adantr 451 . . 3  |-  ( (
ph  /\  ( F  e.  ( ( X Full  Y
)  i^i  ( X Faith  Y ) )  /\  ( 1st `  F ) : R -1-1-onto-> S ) )  ->  Y  e.  B )
141 inss1 3389 . . . . . . 7  |-  ( ( X Full  Y )  i^i  ( X Faith  Y ) )  C_  ( X Full  Y )
142 fullfunc 13780 . . . . . . 7  |-  ( X Full 
Y )  C_  ( X  Func  Y )
143141, 142sstri 3188 . . . . . 6  |-  ( ( X Full  Y )  i^i  ( X Faith  Y ) )  C_  ( X  Func  Y )
144 simprl 732 . . . . . 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 3178 . . . . 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 644 . . . 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 451 . . . . 5  |-  ( (
ph  /\  ( F  e.  ( ( X Full  Y
)  i^i  ( X Faith  Y ) )  /\  ( 1st `  F ) : R -1-1-onto-> S ) )  ->  U  e.  V )
148 eqid 2283 . . . . 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 2358 . . . . . 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 203 . . . . 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 733 . . . . 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 13938 . . . 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 4043 . . 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 13668 . 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 805 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 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543    i^i cin 3151    C_ wss 3152   <.cop 3643   class class class wbr 4023    _I cid 4304   `'ccnv 4688   dom cdm 4689    |` cres 4691    o. ccom 4693   Rel wrel 4694   Fun wfun 5249   -->wf 5251   -1-1-onto->wf1o 5254   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   1stc1st 6120   2ndc2nd 6121   Basecbs 13148    Hom chom 13219  compcco 13220   Catccat 13566   Idccid 13567  Sectcsect 13647  Invcinv 13648    Iso ciso 13649    Func cfunc 13728  idfunccidfu 13729    o.func ccofu 13730   Full cful 13776   Faith cfth 13777  CatCatccatc 13926
This theorem is referenced by:  yoniso  14059
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-fz 10783  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-hom 13232  df-cco 13233  df-cat 13570  df-cid 13571  df-sect 13650  df-inv 13651  df-iso 13652  df-func 13732  df-idfu 13733  df-cofu 13734  df-full 13778  df-fth 13779  df-catc 13927
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