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Theorem invfuc 14173
Description: If  V
( x ) is an inverse to  U ( x ) for each  x, and  U is a natural transformation, then  V is also a natural transformation, and they are inverse in the functor category. (Contributed by Mario Carneiro, 28-Jan-2017.)
Hypotheses
Ref Expression
fuciso.q  |-  Q  =  ( C FuncCat  D )
fuciso.b  |-  B  =  ( Base `  C
)
fuciso.n  |-  N  =  ( C Nat  D )
fuciso.f  |-  ( ph  ->  F  e.  ( C 
Func  D ) )
fuciso.g  |-  ( ph  ->  G  e.  ( C 
Func  D ) )
fucinv.i  |-  I  =  (Inv `  Q )
fucinv.j  |-  J  =  (Inv `  D )
invfuc.u  |-  ( ph  ->  U  e.  ( F N G ) )
invfuc.v  |-  ( (
ph  /\  x  e.  B )  ->  ( U `  x )
( ( ( 1st `  F ) `  x
) J ( ( 1st `  G ) `
 x ) ) X )
Assertion
Ref Expression
invfuc  |-  ( ph  ->  U ( F I G ) ( x  e.  B  |->  X ) )
Distinct variable groups:    x, B    x, C    x, D    x, I    x, F    x, G    x, J    x, N    ph, x    x, Q    x, U
Allowed substitution hint:    X( x)

Proof of Theorem invfuc
Dummy variables  y 
f  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 invfuc.u . 2  |-  ( ph  ->  U  e.  ( F N G ) )
2 invfuc.v . . . . . . . 8  |-  ( (
ph  /\  x  e.  B )  ->  ( U `  x )
( ( ( 1st `  F ) `  x
) J ( ( 1st `  G ) `
 x ) ) X )
3 eqid 2438 . . . . . . . . . 10  |-  ( Base `  D )  =  (
Base `  D )
4 fucinv.j . . . . . . . . . 10  |-  J  =  (Inv `  D )
5 fuciso.f . . . . . . . . . . . . 13  |-  ( ph  ->  F  e.  ( C 
Func  D ) )
6 funcrcl 14062 . . . . . . . . . . . . 13  |-  ( F  e.  ( C  Func  D )  ->  ( C  e.  Cat  /\  D  e. 
Cat ) )
75, 6syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  ( C  e.  Cat  /\  D  e.  Cat )
)
87simprd 451 . . . . . . . . . . 11  |-  ( ph  ->  D  e.  Cat )
98adantr 453 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  B )  ->  D  e.  Cat )
10 fuciso.b . . . . . . . . . . . 12  |-  B  =  ( Base `  C
)
11 relfunc 14061 . . . . . . . . . . . . 13  |-  Rel  ( C  Func  D )
12 1st2ndbr 6398 . . . . . . . . . . . . 13  |-  ( ( Rel  ( C  Func  D )  /\  F  e.  ( C  Func  D
) )  ->  ( 1st `  F ) ( C  Func  D )
( 2nd `  F
) )
1311, 5, 12sylancr 646 . . . . . . . . . . . 12  |-  ( ph  ->  ( 1st `  F
) ( C  Func  D ) ( 2nd `  F
) )
1410, 3, 13funcf1 14065 . . . . . . . . . . 11  |-  ( ph  ->  ( 1st `  F
) : B --> ( Base `  D ) )
1514ffvelrnda 5872 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  B )  ->  (
( 1st `  F
) `  x )  e.  ( Base `  D
) )
16 fuciso.g . . . . . . . . . . . . 13  |-  ( ph  ->  G  e.  ( C 
Func  D ) )
17 1st2ndbr 6398 . . . . . . . . . . . . 13  |-  ( ( Rel  ( C  Func  D )  /\  G  e.  ( C  Func  D
) )  ->  ( 1st `  G ) ( C  Func  D )
( 2nd `  G
) )
1811, 16, 17sylancr 646 . . . . . . . . . . . 12  |-  ( ph  ->  ( 1st `  G
) ( C  Func  D ) ( 2nd `  G
) )
1910, 3, 18funcf1 14065 . . . . . . . . . . 11  |-  ( ph  ->  ( 1st `  G
) : B --> ( Base `  D ) )
2019ffvelrnda 5872 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  B )  ->  (
( 1st `  G
) `  x )  e.  ( Base `  D
) )
21 eqid 2438 . . . . . . . . . 10  |-  (  Hom  `  D )  =  (  Hom  `  D )
223, 4, 9, 15, 20, 21invss 13988 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  B )  ->  (
( ( 1st `  F
) `  x ) J ( ( 1st `  G ) `  x
) )  C_  (
( ( ( 1st `  F ) `  x
) (  Hom  `  D
) ( ( 1st `  G ) `  x
) )  X.  (
( ( 1st `  G
) `  x )
(  Hom  `  D ) ( ( 1st `  F
) `  x )
) ) )
2322ssbrd 4255 . . . . . . . 8  |-  ( (
ph  /\  x  e.  B )  ->  (
( U `  x
) ( ( ( 1st `  F ) `
 x ) J ( ( 1st `  G
) `  x )
) X  ->  ( U `  x )
( ( ( ( 1st `  F ) `
 x ) (  Hom  `  D )
( ( 1st `  G
) `  x )
)  X.  ( ( ( 1st `  G
) `  x )
(  Hom  `  D ) ( ( 1st `  F
) `  x )
) ) X ) )
242, 23mpd 15 . . . . . . 7  |-  ( (
ph  /\  x  e.  B )  ->  ( U `  x )
( ( ( ( 1st `  F ) `
 x ) (  Hom  `  D )
( ( 1st `  G
) `  x )
)  X.  ( ( ( 1st `  G
) `  x )
(  Hom  `  D ) ( ( 1st `  F
) `  x )
) ) X )
25 brxp 4911 . . . . . . . 8  |-  ( ( U `  x ) ( ( ( ( 1st `  F ) `
 x ) (  Hom  `  D )
( ( 1st `  G
) `  x )
)  X.  ( ( ( 1st `  G
) `  x )
(  Hom  `  D ) ( ( 1st `  F
) `  x )
) ) X  <->  ( ( U `  x )  e.  ( ( ( 1st `  F ) `  x
) (  Hom  `  D
) ( ( 1st `  G ) `  x
) )  /\  X  e.  ( ( ( 1st `  G ) `  x
) (  Hom  `  D
) ( ( 1st `  F ) `  x
) ) ) )
2625simprbi 452 . . . . . . 7  |-  ( ( U `  x ) ( ( ( ( 1st `  F ) `
 x ) (  Hom  `  D )
( ( 1st `  G
) `  x )
)  X.  ( ( ( 1st `  G
) `  x )
(  Hom  `  D ) ( ( 1st `  F
) `  x )
) ) X  ->  X  e.  ( (
( 1st `  G
) `  x )
(  Hom  `  D ) ( ( 1st `  F
) `  x )
) )
2724, 26syl 16 . . . . . 6  |-  ( (
ph  /\  x  e.  B )  ->  X  e.  ( ( ( 1st `  G ) `  x
) (  Hom  `  D
) ( ( 1st `  F ) `  x
) ) )
2827ralrimiva 2791 . . . . 5  |-  ( ph  ->  A. x  e.  B  X  e.  ( (
( 1st `  G
) `  x )
(  Hom  `  D ) ( ( 1st `  F
) `  x )
) )
29 fvex 5744 . . . . . . 7  |-  ( Base `  C )  e.  _V
3010, 29eqeltri 2508 . . . . . 6  |-  B  e. 
_V
31 mptelixpg 7101 . . . . . 6  |-  ( B  e.  _V  ->  (
( x  e.  B  |->  X )  e.  X_ x  e.  B  (
( ( 1st `  G
) `  x )
(  Hom  `  D ) ( ( 1st `  F
) `  x )
)  <->  A. x  e.  B  X  e.  ( (
( 1st `  G
) `  x )
(  Hom  `  D ) ( ( 1st `  F
) `  x )
) ) )
3230, 31ax-mp 8 . . . . 5  |-  ( ( x  e.  B  |->  X )  e.  X_ x  e.  B  ( (
( 1st `  G
) `  x )
(  Hom  `  D ) ( ( 1st `  F
) `  x )
)  <->  A. x  e.  B  X  e.  ( (
( 1st `  G
) `  x )
(  Hom  `  D ) ( ( 1st `  F
) `  x )
) )
3328, 32sylibr 205 . . . 4  |-  ( ph  ->  ( x  e.  B  |->  X )  e.  X_ x  e.  B  (
( ( 1st `  G
) `  x )
(  Hom  `  D ) ( ( 1st `  F
) `  x )
) )
34 fveq2 5730 . . . . . 6  |-  ( x  =  y  ->  (
( 1st `  G
) `  x )  =  ( ( 1st `  G ) `  y
) )
35 fveq2 5730 . . . . . 6  |-  ( x  =  y  ->  (
( 1st `  F
) `  x )  =  ( ( 1st `  F ) `  y
) )
3634, 35oveq12d 6101 . . . . 5  |-  ( x  =  y  ->  (
( ( 1st `  G
) `  x )
(  Hom  `  D ) ( ( 1st `  F
) `  x )
)  =  ( ( ( 1st `  G
) `  y )
(  Hom  `  D ) ( ( 1st `  F
) `  y )
) )
3736cbvixpv 7082 . . . 4  |-  X_ x  e.  B  ( (
( 1st `  G
) `  x )
(  Hom  `  D ) ( ( 1st `  F
) `  x )
)  =  X_ y  e.  B  ( (
( 1st `  G
) `  y )
(  Hom  `  D ) ( ( 1st `  F
) `  y )
)
3833, 37syl6eleq 2528 . . 3  |-  ( ph  ->  ( x  e.  B  |->  X )  e.  X_ y  e.  B  (
( ( 1st `  G
) `  y )
(  Hom  `  D ) ( ( 1st `  F
) `  y )
) )
39 simpr2 965 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( y  e.  B  /\  z  e.  B  /\  f  e.  ( y (  Hom  `  C ) z ) ) )  ->  z  e.  B )
40 simpr 449 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  x  e.  B )  ->  x  e.  B )
41 eqid 2438 . . . . . . . . . . . . . . . . . . 19  |-  ( x  e.  B  |->  X )  =  ( x  e.  B  |->  X )
4241fvmpt2 5814 . . . . . . . . . . . . . . . . . 18  |-  ( ( x  e.  B  /\  X  e.  ( (
( 1st `  G
) `  x )
(  Hom  `  D ) ( ( 1st `  F
) `  x )
) )  ->  (
( x  e.  B  |->  X ) `  x
)  =  X )
4340, 27, 42syl2anc 644 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  x  e.  B )  ->  (
( x  e.  B  |->  X ) `  x
)  =  X )
442, 43breqtrrd 4240 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  x  e.  B )  ->  ( U `  x )
( ( ( 1st `  F ) `  x
) J ( ( 1st `  G ) `
 x ) ) ( ( x  e.  B  |->  X ) `  x ) )
4544ralrimiva 2791 . . . . . . . . . . . . . . 15  |-  ( ph  ->  A. x  e.  B  ( U `  x ) ( ( ( 1st `  F ) `  x
) J ( ( 1st `  G ) `
 x ) ) ( ( x  e.  B  |->  X ) `  x ) )
4645adantr 453 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( y  e.  B  /\  z  e.  B  /\  f  e.  ( y (  Hom  `  C ) z ) ) )  ->  A. x  e.  B  ( U `  x ) ( ( ( 1st `  F
) `  x ) J ( ( 1st `  G ) `  x
) ) ( ( x  e.  B  |->  X ) `  x ) )
47 nfcv 2574 . . . . . . . . . . . . . . . 16  |-  F/_ x
( U `  z
)
48 nfcv 2574 . . . . . . . . . . . . . . . 16  |-  F/_ x
( ( ( 1st `  F ) `  z
) J ( ( 1st `  G ) `
 z ) )
49 nffvmpt1 5738 . . . . . . . . . . . . . . . 16  |-  F/_ x
( ( x  e.  B  |->  X ) `  z )
5047, 48, 49nfbr 4258 . . . . . . . . . . . . . . 15  |-  F/ x
( U `  z
) ( ( ( 1st `  F ) `
 z ) J ( ( 1st `  G
) `  z )
) ( ( x  e.  B  |->  X ) `
 z )
51 fveq2 5730 . . . . . . . . . . . . . . . 16  |-  ( x  =  z  ->  ( U `  x )  =  ( U `  z ) )
52 fveq2 5730 . . . . . . . . . . . . . . . . 17  |-  ( x  =  z  ->  (
( 1st `  F
) `  x )  =  ( ( 1st `  F ) `  z
) )
53 fveq2 5730 . . . . . . . . . . . . . . . . 17  |-  ( x  =  z  ->  (
( 1st `  G
) `  x )  =  ( ( 1st `  G ) `  z
) )
5452, 53oveq12d 6101 . . . . . . . . . . . . . . . 16  |-  ( x  =  z  ->  (
( ( 1st `  F
) `  x ) J ( ( 1st `  G ) `  x
) )  =  ( ( ( 1st `  F
) `  z ) J ( ( 1st `  G ) `  z
) ) )
55 fveq2 5730 . . . . . . . . . . . . . . . 16  |-  ( x  =  z  ->  (
( x  e.  B  |->  X ) `  x
)  =  ( ( x  e.  B  |->  X ) `  z ) )
5651, 54, 55breq123d 4228 . . . . . . . . . . . . . . 15  |-  ( x  =  z  ->  (
( U `  x
) ( ( ( 1st `  F ) `
 x ) J ( ( 1st `  G
) `  x )
) ( ( x  e.  B  |->  X ) `
 x )  <->  ( U `  z ) ( ( ( 1st `  F
) `  z ) J ( ( 1st `  G ) `  z
) ) ( ( x  e.  B  |->  X ) `  z ) ) )
5750, 56rspc 3048 . . . . . . . . . . . . . 14  |-  ( z  e.  B  ->  ( A. x  e.  B  ( U `  x ) ( ( ( 1st `  F ) `  x
) J ( ( 1st `  G ) `
 x ) ) ( ( x  e.  B  |->  X ) `  x )  ->  ( U `  z )
( ( ( 1st `  F ) `  z
) J ( ( 1st `  G ) `
 z ) ) ( ( x  e.  B  |->  X ) `  z ) ) )
5839, 46, 57sylc 59 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( y  e.  B  /\  z  e.  B  /\  f  e.  ( y (  Hom  `  C ) z ) ) )  ->  ( U `  z )
( ( ( 1st `  F ) `  z
) J ( ( 1st `  G ) `
 z ) ) ( ( x  e.  B  |->  X ) `  z ) )
598adantr 453 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( y  e.  B  /\  z  e.  B  /\  f  e.  ( y (  Hom  `  C ) z ) ) )  ->  D  e.  Cat )
6014adantr 453 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( y  e.  B  /\  z  e.  B  /\  f  e.  ( y (  Hom  `  C ) z ) ) )  ->  ( 1st `  F ) : B --> ( Base `  D
) )
6160, 39ffvelrnd 5873 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( y  e.  B  /\  z  e.  B  /\  f  e.  ( y (  Hom  `  C ) z ) ) )  ->  (
( 1st `  F
) `  z )  e.  ( Base `  D
) )
6219adantr 453 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( y  e.  B  /\  z  e.  B  /\  f  e.  ( y (  Hom  `  C ) z ) ) )  ->  ( 1st `  G ) : B --> ( Base `  D
) )
6362, 39ffvelrnd 5873 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( y  e.  B  /\  z  e.  B  /\  f  e.  ( y (  Hom  `  C ) z ) ) )  ->  (
( 1st `  G
) `  z )  e.  ( Base `  D
) )
64 eqid 2438 . . . . . . . . . . . . . 14  |-  (Sect `  D )  =  (Sect `  D )
653, 4, 59, 61, 63, 64isinv 13987 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( y  e.  B  /\  z  e.  B  /\  f  e.  ( y (  Hom  `  C ) z ) ) )  ->  (
( U `  z
) ( ( ( 1st `  F ) `
 z ) J ( ( 1st `  G
) `  z )
) ( ( x  e.  B  |->  X ) `
 z )  <->  ( ( U `  z )
( ( ( 1st `  F ) `  z
) (Sect `  D
) ( ( 1st `  G ) `  z
) ) ( ( x  e.  B  |->  X ) `  z )  /\  ( ( x  e.  B  |->  X ) `
 z ) ( ( ( 1st `  G
) `  z )
(Sect `  D )
( ( 1st `  F
) `  z )
) ( U `  z ) ) ) )
6658, 65mpbid 203 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( y  e.  B  /\  z  e.  B  /\  f  e.  ( y (  Hom  `  C ) z ) ) )  ->  (
( U `  z
) ( ( ( 1st `  F ) `
 z ) (Sect `  D ) ( ( 1st `  G ) `
 z ) ) ( ( x  e.  B  |->  X ) `  z )  /\  (
( x  e.  B  |->  X ) `  z
) ( ( ( 1st `  G ) `
 z ) (Sect `  D ) ( ( 1st `  F ) `
 z ) ) ( U `  z
) ) )
6766simpld 447 . . . . . . . . . . 11  |-  ( (
ph  /\  ( y  e.  B  /\  z  e.  B  /\  f  e.  ( y (  Hom  `  C ) z ) ) )  ->  ( U `  z )
( ( ( 1st `  F ) `  z
) (Sect `  D
) ( ( 1st `  G ) `  z
) ) ( ( x  e.  B  |->  X ) `  z ) )
68 eqid 2438 . . . . . . . . . . . 12  |-  (comp `  D )  =  (comp `  D )
69 eqid 2438 . . . . . . . . . . . 12  |-  ( Id
`  D )  =  ( Id `  D
)
703, 21, 68, 69, 64, 59, 61, 63issect 13981 . . . . . . . . . . 11  |-  ( (
ph  /\  ( y  e.  B  /\  z  e.  B  /\  f  e.  ( y (  Hom  `  C ) z ) ) )  ->  (
( U `  z
) ( ( ( 1st `  F ) `
 z ) (Sect `  D ) ( ( 1st `  G ) `
 z ) ) ( ( x  e.  B  |->  X ) `  z )  <->  ( ( U `  z )  e.  ( ( ( 1st `  F ) `  z
) (  Hom  `  D
) ( ( 1st `  G ) `  z
) )  /\  (
( x  e.  B  |->  X ) `  z
)  e.  ( ( ( 1st `  G
) `  z )
(  Hom  `  D ) ( ( 1st `  F
) `  z )
)  /\  ( (
( x  e.  B  |->  X ) `  z
) ( <. (
( 1st `  F
) `  z ) ,  ( ( 1st `  G ) `  z
) >. (comp `  D
) ( ( 1st `  F ) `  z
) ) ( U `
 z ) )  =  ( ( Id
`  D ) `  ( ( 1st `  F
) `  z )
) ) ) )
7167, 70mpbid 203 . . . . . . . . . 10  |-  ( (
ph  /\  ( y  e.  B  /\  z  e.  B  /\  f  e.  ( y (  Hom  `  C ) z ) ) )  ->  (
( U `  z
)  e.  ( ( ( 1st `  F
) `  z )
(  Hom  `  D ) ( ( 1st `  G
) `  z )
)  /\  ( (
x  e.  B  |->  X ) `  z )  e.  ( ( ( 1st `  G ) `
 z ) (  Hom  `  D )
( ( 1st `  F
) `  z )
)  /\  ( (
( x  e.  B  |->  X ) `  z
) ( <. (
( 1st `  F
) `  z ) ,  ( ( 1st `  G ) `  z
) >. (comp `  D
) ( ( 1st `  F ) `  z
) ) ( U `
 z ) )  =  ( ( Id
`  D ) `  ( ( 1st `  F
) `  z )
) ) )
7271simp3d 972 . . . . . . . . 9  |-  ( (
ph  /\  ( y  e.  B  /\  z  e.  B  /\  f  e.  ( y (  Hom  `  C ) z ) ) )  ->  (
( ( x  e.  B  |->  X ) `  z ) ( <.
( ( 1st `  F
) `  z ) ,  ( ( 1st `  G ) `  z
) >. (comp `  D
) ( ( 1st `  F ) `  z
) ) ( U `
 z ) )  =  ( ( Id
`  D ) `  ( ( 1st `  F
) `  z )
) )
7372oveq1d 6098 . . . . . . . 8  |-  ( (
ph  /\  ( y  e.  B  /\  z  e.  B  /\  f  e.  ( y (  Hom  `  C ) z ) ) )  ->  (
( ( ( x  e.  B  |->  X ) `
 z ) (
<. ( ( 1st `  F
) `  z ) ,  ( ( 1st `  G ) `  z
) >. (comp `  D
) ( ( 1st `  F ) `  z
) ) ( U `
 z ) ) ( <. ( ( 1st `  F ) `  y
) ,  ( ( 1st `  F ) `
 z ) >.
(comp `  D )
( ( 1st `  F
) `  z )
) ( ( y ( 2nd `  F
) z ) `  f ) )  =  ( ( ( Id
`  D ) `  ( ( 1st `  F
) `  z )
) ( <. (
( 1st `  F
) `  y ) ,  ( ( 1st `  F ) `  z
) >. (comp `  D
) ( ( 1st `  F ) `  z
) ) ( ( y ( 2nd `  F
) z ) `  f ) ) )
74 simpr1 964 . . . . . . . . . 10  |-  ( (
ph  /\  ( y  e.  B  /\  z  e.  B  /\  f  e.  ( y (  Hom  `  C ) z ) ) )  ->  y  e.  B )
7560, 74ffvelrnd 5873 . . . . . . . . 9  |-  ( (
ph  /\  ( y  e.  B  /\  z  e.  B  /\  f  e.  ( y (  Hom  `  C ) z ) ) )  ->  (
( 1st `  F
) `  y )  e.  ( Base `  D
) )
76 eqid 2438 . . . . . . . . . . 11  |-  (  Hom  `  C )  =  (  Hom  `  C )
7713adantr 453 . . . . . . . . . . 11  |-  ( (
ph  /\  ( y  e.  B  /\  z  e.  B  /\  f  e.  ( y (  Hom  `  C ) z ) ) )  ->  ( 1st `  F ) ( C  Func  D )
( 2nd `  F
) )
7810, 76, 21, 77, 74, 39funcf2 14067 . . . . . . . . . 10  |-  ( (
ph  /\  ( y  e.  B  /\  z  e.  B  /\  f  e.  ( y (  Hom  `  C ) z ) ) )  ->  (
y ( 2nd `  F
) z ) : ( y (  Hom  `  C ) z ) --> ( ( ( 1st `  F ) `  y
) (  Hom  `  D
) ( ( 1st `  F ) `  z
) ) )
79 simpr3 966 . . . . . . . . . 10  |-  ( (
ph  /\  ( y  e.  B  /\  z  e.  B  /\  f  e.  ( y (  Hom  `  C ) z ) ) )  ->  f  e.  ( y (  Hom  `  C ) z ) )
8078, 79ffvelrnd 5873 . . . . . . . . 9  |-  ( (
ph  /\  ( y  e.  B  /\  z  e.  B  /\  f  e.  ( y (  Hom  `  C ) z ) ) )  ->  (
( y ( 2nd `  F ) z ) `
 f )  e.  ( ( ( 1st `  F ) `  y
) (  Hom  `  D
) ( ( 1st `  F ) `  z
) ) )
813, 21, 69, 59, 75, 68, 61, 80catlid 13910 . . . . . . . 8  |-  ( (
ph  /\  ( y  e.  B  /\  z  e.  B  /\  f  e.  ( y (  Hom  `  C ) z ) ) )  ->  (
( ( Id `  D ) `  (
( 1st `  F
) `  z )
) ( <. (
( 1st `  F
) `  y ) ,  ( ( 1st `  F ) `  z
) >. (comp `  D
) ( ( 1st `  F ) `  z
) ) ( ( y ( 2nd `  F
) z ) `  f ) )  =  ( ( y ( 2nd `  F ) z ) `  f
) )
8273, 81eqtr2d 2471 . . . . . . 7  |-  ( (
ph  /\  ( y  e.  B  /\  z  e.  B  /\  f  e.  ( y (  Hom  `  C ) z ) ) )  ->  (
( y ( 2nd `  F ) z ) `
 f )  =  ( ( ( ( x  e.  B  |->  X ) `  z ) ( <. ( ( 1st `  F ) `  z
) ,  ( ( 1st `  G ) `
 z ) >.
(comp `  D )
( ( 1st `  F
) `  z )
) ( U `  z ) ) (
<. ( ( 1st `  F
) `  y ) ,  ( ( 1st `  F ) `  z
) >. (comp `  D
) ( ( 1st `  F ) `  z
) ) ( ( y ( 2nd `  F
) z ) `  f ) ) )
83 fuciso.n . . . . . . . . 9  |-  N  =  ( C Nat  D )
841adantr 453 . . . . . . . . . 10  |-  ( (
ph  /\  ( y  e.  B  /\  z  e.  B  /\  f  e.  ( y (  Hom  `  C ) z ) ) )  ->  U  e.  ( F N G ) )
8583, 84nat1st2nd 14150 . . . . . . . . 9  |-  ( (
ph  /\  ( y  e.  B  /\  z  e.  B  /\  f  e.  ( y (  Hom  `  C ) z ) ) )  ->  U  e.  ( <. ( 1st `  F
) ,  ( 2nd `  F ) >. N <. ( 1st `  G ) ,  ( 2nd `  G
) >. ) )
8683, 85, 10, 21, 39natcl 14152 . . . . . . . 8  |-  ( (
ph  /\  ( y  e.  B  /\  z  e.  B  /\  f  e.  ( y (  Hom  `  C ) z ) ) )  ->  ( U `  z )  e.  ( ( ( 1st `  F ) `  z
) (  Hom  `  D
) ( ( 1st `  G ) `  z
) ) )
8771simp2d 971 . . . . . . . 8  |-  ( (
ph  /\  ( y  e.  B  /\  z  e.  B  /\  f  e.  ( y (  Hom  `  C ) z ) ) )  ->  (
( x  e.  B  |->  X ) `  z
)  e.  ( ( ( 1st `  G
) `  z )
(  Hom  `  D ) ( ( 1st `  F
) `  z )
) )
883, 21, 68, 59, 75, 61, 63, 80, 86, 61, 87catass 13913 . . . . . . 7  |-  ( (
ph  /\  ( y  e.  B  /\  z  e.  B  /\  f  e.  ( y (  Hom  `  C ) z ) ) )  ->  (
( ( ( x  e.  B  |->  X ) `
 z ) (
<. ( ( 1st `  F
) `  z ) ,  ( ( 1st `  G ) `  z
) >. (comp `  D
) ( ( 1st `  F ) `  z
) ) ( U `
 z ) ) ( <. ( ( 1st `  F ) `  y
) ,  ( ( 1st `  F ) `
 z ) >.
(comp `  D )
( ( 1st `  F
) `  z )
) ( ( y ( 2nd `  F
) z ) `  f ) )  =  ( ( ( x  e.  B  |->  X ) `
 z ) (
<. ( ( 1st `  F
) `  y ) ,  ( ( 1st `  G ) `  z
) >. (comp `  D
) ( ( 1st `  F ) `  z
) ) ( ( U `  z ) ( <. ( ( 1st `  F ) `  y
) ,  ( ( 1st `  F ) `
 z ) >.
(comp `  D )
( ( 1st `  G
) `  z )
) ( ( y ( 2nd `  F
) z ) `  f ) ) ) )
8983, 85, 10, 76, 68, 74, 39, 79nati 14154 . . . . . . . 8  |-  ( (
ph  /\  ( y  e.  B  /\  z  e.  B  /\  f  e.  ( y (  Hom  `  C ) z ) ) )  ->  (
( U `  z
) ( <. (
( 1st `  F
) `  y ) ,  ( ( 1st `  F ) `  z
) >. (comp `  D
) ( ( 1st `  G ) `  z
) ) ( ( y ( 2nd `  F
) z ) `  f ) )  =  ( ( ( y ( 2nd `  G
) z ) `  f ) ( <.
( ( 1st `  F
) `  y ) ,  ( ( 1st `  G ) `  y
) >. (comp `  D
) ( ( 1st `  G ) `  z
) ) ( U `
 y ) ) )
9089oveq2d 6099 . . . . . . 7  |-  ( (
ph  /\  ( y  e.  B  /\  z  e.  B  /\  f  e.  ( y (  Hom  `  C ) z ) ) )  ->  (
( ( x  e.  B  |->  X ) `  z ) ( <.
( ( 1st `  F
) `  y ) ,  ( ( 1st `  G ) `  z
) >. (comp `  D
) ( ( 1st `  F ) `  z
) ) ( ( U `  z ) ( <. ( ( 1st `  F ) `  y
) ,  ( ( 1st `  F ) `
 z ) >.
(comp `  D )
( ( 1st `  G
) `  z )
) ( ( y ( 2nd `  F
) z ) `  f ) ) )  =  ( ( ( x  e.  B  |->  X ) `  z ) ( <. ( ( 1st `  F ) `  y
) ,  ( ( 1st `  G ) `
 z ) >.
(comp `  D )
( ( 1st `  F
) `  z )
) ( ( ( y ( 2nd `  G
) z ) `  f ) ( <.
( ( 1st `  F
) `  y ) ,  ( ( 1st `  G ) `  y
) >. (comp `  D
) ( ( 1st `  G ) `  z
) ) ( U `
 y ) ) ) )
9182, 88, 903eqtrd 2474 . . . . . 6  |-  ( (
ph  /\  ( y  e.  B  /\  z  e.  B  /\  f  e.  ( y (  Hom  `  C ) z ) ) )  ->  (
( y ( 2nd `  F ) z ) `
 f )  =  ( ( ( x  e.  B  |->  X ) `
 z ) (
<. ( ( 1st `  F
) `  y ) ,  ( ( 1st `  G ) `  z
) >. (comp `  D
) ( ( 1st `  F ) `  z
) ) ( ( ( y ( 2nd `  G ) z ) `
 f ) (
<. ( ( 1st `  F
) `  y ) ,  ( ( 1st `  G ) `  y
) >. (comp `  D
) ( ( 1st `  G ) `  z
) ) ( U `
 y ) ) ) )
9291oveq1d 6098 . . . . 5  |-  ( (
ph  /\  ( y  e.  B  /\  z  e.  B  /\  f  e.  ( y (  Hom  `  C ) z ) ) )  ->  (
( ( y ( 2nd `  F ) z ) `  f
) ( <. (
( 1st `  G
) `  y ) ,  ( ( 1st `  F ) `  y
) >. (comp `  D
) ( ( 1st `  F ) `  z
) ) ( ( x  e.  B  |->  X ) `  y ) )  =  ( ( ( ( x  e.  B  |->  X ) `  z ) ( <.
( ( 1st `  F
) `  y ) ,  ( ( 1st `  G ) `  z
) >. (comp `  D
) ( ( 1st `  F ) `  z
) ) ( ( ( y ( 2nd `  G ) z ) `
 f ) (
<. ( ( 1st `  F
) `  y ) ,  ( ( 1st `  G ) `  y
) >. (comp `  D
) ( ( 1st `  G ) `  z
) ) ( U `
 y ) ) ) ( <. (
( 1st `  G
) `  y ) ,  ( ( 1st `  F ) `  y
) >. (comp `  D
) ( ( 1st `  F ) `  z
) ) ( ( x  e.  B  |->  X ) `  y ) ) )
9362, 74ffvelrnd 5873 . . . . . 6  |-  ( (
ph  /\  ( y  e.  B  /\  z  e.  B  /\  f  e.  ( y (  Hom  `  C ) z ) ) )  ->  (
( 1st `  G
) `  y )  e.  ( Base `  D
) )
94 nfcv 2574 . . . . . . . . . . . . 13  |-  F/_ x
( U `  y
)
95 nfcv 2574 . . . . . . . . . . . . 13  |-  F/_ x
( ( ( 1st `  F ) `  y
) J ( ( 1st `  G ) `
 y ) )
96 nffvmpt1 5738 . . . . . . . . . . . . 13  |-  F/_ x
( ( x  e.  B  |->  X ) `  y )
9794, 95, 96nfbr 4258 . . . . . . . . . . . 12  |-  F/ x
( U `  y
) ( ( ( 1st `  F ) `
 y ) J ( ( 1st `  G
) `  y )
) ( ( x  e.  B  |->  X ) `
 y )
98 fveq2 5730 . . . . . . . . . . . . 13  |-  ( x  =  y  ->  ( U `  x )  =  ( U `  y ) )
9935, 34oveq12d 6101 . . . . . . . . . . . . 13  |-  ( x  =  y  ->  (
( ( 1st `  F
) `  x ) J ( ( 1st `  G ) `  x
) )  =  ( ( ( 1st `  F
) `  y ) J ( ( 1st `  G ) `  y
) ) )
100 fveq2 5730 . . . . . . . . . . . . 13  |-  ( x  =  y  ->  (
( x  e.  B  |->  X ) `  x
)  =  ( ( x  e.  B  |->  X ) `  y ) )
10198, 99, 100breq123d 4228 . . . . . . . . . . . 12  |-  ( x  =  y  ->  (
( U `  x
) ( ( ( 1st `  F ) `
 x ) J ( ( 1st `  G
) `  x )
) ( ( x  e.  B  |->  X ) `
 x )  <->  ( U `  y ) ( ( ( 1st `  F
) `  y ) J ( ( 1st `  G ) `  y
) ) ( ( x  e.  B  |->  X ) `  y ) ) )
10297, 101rspc 3048 . . . . . . . . . . 11  |-  ( y  e.  B  ->  ( A. x  e.  B  ( U `  x ) ( ( ( 1st `  F ) `  x
) J ( ( 1st `  G ) `
 x ) ) ( ( x  e.  B  |->  X ) `  x )  ->  ( U `  y )
( ( ( 1st `  F ) `  y
) J ( ( 1st `  G ) `
 y ) ) ( ( x  e.  B  |->  X ) `  y ) ) )
10374, 46, 102sylc 59 . . . . . . . . . 10  |-  ( (
ph  /\  ( y  e.  B  /\  z  e.  B  /\  f  e.  ( y (  Hom  `  C ) z ) ) )  ->  ( U `  y )
( ( ( 1st `  F ) `  y
) J ( ( 1st `  G ) `
 y ) ) ( ( x  e.  B  |->  X ) `  y ) )
1043, 4, 59, 75, 93, 64isinv 13987 . . . . . . . . . 10  |-  ( (
ph  /\  ( y  e.  B  /\  z  e.  B  /\  f  e.  ( y (  Hom  `  C ) z ) ) )  ->  (
( U `  y
) ( ( ( 1st `  F ) `
 y ) J ( ( 1st `  G
) `  y )
) ( ( x  e.  B  |->  X ) `
 y )  <->  ( ( U `  y )
( ( ( 1st `  F ) `  y
) (Sect `  D
) ( ( 1st `  G ) `  y
) ) ( ( x  e.  B  |->  X ) `  y )  /\  ( ( x  e.  B  |->  X ) `
 y ) ( ( ( 1st `  G
) `  y )
(Sect `  D )
( ( 1st `  F
) `  y )
) ( U `  y ) ) ) )
105103, 104mpbid 203 . . . . . . . . 9  |-  ( (
ph  /\  ( y  e.  B  /\  z  e.  B  /\  f  e.  ( y (  Hom  `  C ) z ) ) )  ->  (
( U `  y
) ( ( ( 1st `  F ) `
 y ) (Sect `  D ) ( ( 1st `  G ) `
 y ) ) ( ( x  e.  B  |->  X ) `  y )  /\  (
( x  e.  B  |->  X ) `  y
) ( ( ( 1st `  G ) `
 y ) (Sect `  D ) ( ( 1st `  F ) `
 y ) ) ( U `  y
) ) )
106105simprd 451 . . . . . . . 8  |-  ( (
ph  /\  ( y  e.  B  /\  z  e.  B  /\  f  e.  ( y (  Hom  `  C ) z ) ) )  ->  (
( x  e.  B  |->  X ) `  y
) ( ( ( 1st `  G ) `
 y ) (Sect `  D ) ( ( 1st `  F ) `
 y ) ) ( U `  y
) )
1073, 21, 68, 69, 64, 59, 93, 75issect 13981 . . . . . . . 8  |-  ( (
ph  /\  ( y  e.  B  /\  z  e.  B  /\  f  e.  ( y (  Hom  `  C ) z ) ) )  ->  (
( ( x  e.  B  |->  X ) `  y ) ( ( ( 1st `  G
) `  y )
(Sect `  D )
( ( 1st `  F
) `  y )
) ( U `  y )  <->  ( (
( x  e.  B  |->  X ) `  y
)  e.  ( ( ( 1st `  G
) `  y )
(  Hom  `  D ) ( ( 1st `  F
) `  y )
)  /\  ( U `  y )  e.  ( ( ( 1st `  F
) `  y )
(  Hom  `  D ) ( ( 1st `  G
) `  y )
)  /\  ( ( U `  y )
( <. ( ( 1st `  G ) `  y
) ,  ( ( 1st `  F ) `
 y ) >.
(comp `  D )
( ( 1st `  G
) `  y )
) ( ( x  e.  B  |->  X ) `
 y ) )  =  ( ( Id
`  D ) `  ( ( 1st `  G
) `  y )
) ) ) )
108106, 107mpbid 203 . . . . . . 7  |-  ( (
ph  /\  ( y  e.  B  /\  z  e.  B  /\  f  e.  ( y (  Hom  `  C ) z ) ) )  ->  (
( ( x  e.  B  |->  X ) `  y )  e.  ( ( ( 1st `  G
) `  y )
(  Hom  `  D ) ( ( 1st `  F
) `  y )
)  /\  ( U `  y )  e.  ( ( ( 1st `  F
) `  y )
(  Hom  `  D ) ( ( 1st `  G
) `  y )
)  /\  ( ( U `  y )
( <. ( ( 1st `  G ) `  y
) ,  ( ( 1st `  F ) `
 y ) >.
(comp `  D )
( ( 1st `  G
) `  y )
) ( ( x  e.  B  |->  X ) `
 y ) )  =  ( ( Id
`  D ) `  ( ( 1st `  G
) `  y )
) ) )
109108simp1d 970 . . . . . 6  |-  ( (
ph  /\  ( y  e.  B  /\  z  e.  B  /\  f  e.  ( y (  Hom  `  C ) z ) ) )  ->  (
( x  e.  B  |->  X ) `  y
)  e.  ( ( ( 1st `  G
) `  y )
(  Hom  `  D ) ( ( 1st `  F
) `  y )
) )
110108simp2d 971 . . . . . . 7  |-  ( (
ph  /\  ( y  e.  B  /\  z  e.  B  /\  f  e.  ( y (  Hom  `  C ) z ) ) )  ->  ( U `  y )  e.  ( ( ( 1st `  F ) `  y
) (  Hom  `  D
) ( ( 1st `  G ) `  y
) ) )
11118adantr 453 . . . . . . . . 9  |-  ( (
ph  /\  ( y  e.  B  /\  z  e.  B  /\  f  e.  ( y (  Hom  `  C ) z ) ) )  ->  ( 1st `  G ) ( C  Func  D )
( 2nd `  G
) )
11210, 76, 21, 111, 74, 39funcf2 14067 . . . . . . . 8  |-  ( (
ph  /\  ( y  e.  B  /\  z  e.  B  /\  f  e.  ( y (  Hom  `  C ) z ) ) )  ->  (
y ( 2nd `  G
) z ) : ( y (  Hom  `  C ) z ) --> ( ( ( 1st `  G ) `  y
) (  Hom  `  D
) ( ( 1st `  G ) `  z
) ) )
113112, 79ffvelrnd 5873 . . . . . . 7  |-  ( (
ph  /\  ( y  e.  B  /\  z  e.  B  /\  f  e.  ( y (  Hom  `  C ) z ) ) )  ->  (
( y ( 2nd `  G ) z ) `
 f )  e.  ( ( ( 1st `  G ) `  y
) (  Hom  `  D
) ( ( 1st `  G ) `  z
) ) )
1143, 21, 68, 59, 75, 93, 63, 110, 113catcocl 13912 . . . . . 6  |-  ( (
ph  /\  ( y  e.  B  /\  z  e.  B  /\  f  e.  ( y (  Hom  `  C ) z ) ) )  ->  (
( ( y ( 2nd `  G ) z ) `  f
) ( <. (
( 1st `  F
) `  y ) ,  ( ( 1st `  G ) `  y
) >. (comp `  D
) ( ( 1st `  G ) `  z
) ) ( U `
 y ) )  e.  ( ( ( 1st `  F ) `
 y ) (  Hom  `  D )
( ( 1st `  G
) `  z )
) )
1153, 21, 68, 59, 93, 75, 63, 109, 114, 61, 87catass 13913 . . . . 5  |-  ( (
ph  /\  ( y  e.  B  /\  z  e.  B  /\  f  e.  ( y (  Hom  `  C ) z ) ) )  ->  (
( ( ( x  e.  B  |->  X ) `
 z ) (
<. ( ( 1st `  F
) `  y ) ,  ( ( 1st `  G ) `  z
) >. (comp `  D
) ( ( 1st `  F ) `  z
) ) ( ( ( y ( 2nd `  G ) z ) `
 f ) (
<. ( ( 1st `  F
) `  y ) ,  ( ( 1st `  G ) `  y
) >. (comp `  D
) ( ( 1st `  G ) `  z
) ) ( U `
 y ) ) ) ( <. (
( 1st `  G
) `  y ) ,  ( ( 1st `  F ) `  y
) >. (comp `  D
) ( ( 1st `  F ) `  z
) ) ( ( x  e.  B  |->  X ) `  y ) )  =  ( ( ( x  e.  B  |->  X ) `  z
) ( <. (
( 1st `  G
) `  y ) ,  ( ( 1st `  G ) `  z
) >. (comp `  D
) ( ( 1st `  F ) `  z
) ) ( ( ( ( y ( 2nd `  G ) z ) `  f
) ( <. (
( 1st `  F
) `  y ) ,  ( ( 1st `  G ) `  y
) >. (comp `  D
) ( ( 1st `  G ) `  z
) ) ( U `
 y ) ) ( <. ( ( 1st `  G ) `  y
) ,  ( ( 1st `  F ) `
 y ) >.
(comp `  D )
( ( 1st `  G
) `  z )
) ( ( x  e.  B  |->  X ) `
 y ) ) ) )
11683, 85, 10, 21, 74natcl 14152 . . . . . . . 8  |-  ( (
ph  /\  ( y  e.  B  /\  z  e.  B  /\  f  e.  ( y (  Hom  `  C ) z ) ) )  ->  ( U `  y )  e.  ( ( ( 1st `  F ) `  y
) (  Hom  `  D
) ( ( 1st `  G ) `  y
) ) )
1173, 21, 68, 59, 93, 75, 93, 109, 116, 63, 113catass 13913 . . . . . . 7  |-  ( (
ph  /\  ( y  e.  B  /\  z  e.  B  /\  f  e.  ( y (  Hom  `  C ) z ) ) )  ->  (
( ( ( y ( 2nd `  G
) z ) `  f ) ( <.
( ( 1st `  F
) `  y ) ,  ( ( 1st `  G ) `  y
) >. (comp `  D
) ( ( 1st `  G ) `  z
) ) ( U `
 y ) ) ( <. ( ( 1st `  G ) `  y
) ,  ( ( 1st `  F ) `
 y ) >.
(comp `  D )
( ( 1st `  G
) `  z )
) ( ( x  e.  B  |->  X ) `
 y ) )  =  ( ( ( y ( 2nd `  G
) z ) `  f ) ( <.
( ( 1st `  G
) `  y ) ,  ( ( 1st `  G ) `  y
) >. (comp `  D
) ( ( 1st `  G ) `  z
) ) ( ( U `  y ) ( <. ( ( 1st `  G ) `  y
) ,  ( ( 1st `  F ) `
 y ) >.
(comp `  D )
( ( 1st `  G
) `  y )
) ( ( x  e.  B  |->  X ) `
 y ) ) ) )
118108simp3d 972 . . . . . . . 8  |-  ( (
ph  /\  ( y  e.  B  /\  z  e.  B  /\  f  e.  ( y (  Hom  `  C ) z ) ) )  ->  (
( U `  y
) ( <. (
( 1st `  G
) `  y ) ,  ( ( 1st `  F ) `  y
) >. (comp `  D
) ( ( 1st `  G ) `  y
) ) ( ( x  e.  B  |->  X ) `  y ) )  =  ( ( Id `  D ) `
 ( ( 1st `  G ) `  y
) ) )
119118oveq2d 6099 . . . . . . 7  |-  ( (
ph  /\  ( y  e.  B  /\  z  e.  B  /\  f  e.  ( y (  Hom  `  C ) z ) ) )  ->  (
( ( y ( 2nd `  G ) z ) `  f
) ( <. (
( 1st `  G
) `  y ) ,  ( ( 1st `  G ) `  y
) >. (comp `  D
) ( ( 1st `  G ) `  z
) ) ( ( U `  y ) ( <. ( ( 1st `  G ) `  y
) ,  ( ( 1st `  F ) `
 y ) >.
(comp `  D )
( ( 1st `  G
) `  y )
) ( ( x  e.  B  |->  X ) `
 y ) ) )  =  ( ( ( y ( 2nd `  G ) z ) `
 f ) (
<. ( ( 1st `  G
) `  y ) ,  ( ( 1st `  G ) `  y
) >. (comp `  D
) ( ( 1st `  G ) `  z
) ) ( ( Id `  D ) `
 ( ( 1st `  G ) `  y
) ) ) )
1203, 21, 69, 59, 93, 68, 63, 113catrid 13911 . . . . . . 7  |-  ( (
ph  /\  ( y  e.  B  /\  z  e.  B  /\  f  e.  ( y (  Hom  `  C ) z ) ) )  ->  (
( ( y ( 2nd `  G ) z ) `  f
) ( <. (
( 1st `  G
) `  y ) ,  ( ( 1st `  G ) `  y
) >. (comp `  D
) ( ( 1st `  G ) `  z
) ) ( ( Id `  D ) `
 ( ( 1st `  G ) `  y
) ) )  =  ( ( y ( 2nd `  G ) z ) `  f
) )
121117, 119, 1203eqtrd 2474 . . . . . 6  |-  ( (
ph  /\  ( y  e.  B  /\  z  e.  B  /\  f  e.  ( y (  Hom  `  C ) z ) ) )  ->  (
( ( ( y ( 2nd `  G
) z ) `  f ) ( <.
( ( 1st `  F
) `  y ) ,  ( ( 1st `  G ) `  y
) >. (comp `  D
) ( ( 1st `  G ) `  z
) ) ( U `
 y ) ) ( <. ( ( 1st `  G ) `  y
) ,  ( ( 1st `  F ) `
 y ) >.
(comp `  D )
( ( 1st `  G
) `  z )
) ( ( x  e.  B  |->  X ) `
 y ) )  =  ( ( y ( 2nd `  G
) z ) `  f ) )
122121oveq2d 6099 . . . . 5  |-  ( (
ph  /\  ( y  e.  B  /\  z  e.  B  /\  f  e.  ( y (  Hom  `  C ) z ) ) )  ->  (
( ( x  e.  B  |->  X ) `  z ) ( <.
( ( 1st `  G
) `  y ) ,  ( ( 1st `  G ) `  z
) >. (comp `  D
) ( ( 1st `  F ) `  z
) ) ( ( ( ( y ( 2nd `  G ) z ) `  f
) ( <. (
( 1st `  F
) `  y ) ,  ( ( 1st `  G ) `  y
) >. (comp `  D
) ( ( 1st `  G ) `  z
) ) ( U `
 y ) ) ( <. ( ( 1st `  G ) `  y
) ,  ( ( 1st `  F ) `
 y ) >.
(comp `  D )
( ( 1st `  G
) `  z )
) ( ( x  e.  B  |->  X ) `
 y ) ) )  =  ( ( ( x  e.  B  |->  X ) `  z
) ( <. (
( 1st `  G
) `  y ) ,  ( ( 1st `  G ) `  z
) >. (comp `  D
) ( ( 1st `  F ) `  z
) ) ( ( y ( 2nd `  G
) z ) `  f ) ) )
12392, 115, 1223eqtrrd 2475 . . . 4  |-  ( (
ph  /\  ( y  e.  B  /\  z  e.  B  /\  f  e.  ( y (  Hom  `  C ) z ) ) )  ->  (
( ( x  e.  B  |->  X ) `  z ) ( <.
( ( 1st `  G
) `  y ) ,  ( ( 1st `  G ) `  z
) >. (comp `  D
) ( ( 1st `  F ) `  z
) ) ( ( y ( 2nd `  G
) z ) `  f ) )  =  ( ( ( y ( 2nd `  F
) z ) `  f ) ( <.
( ( 1st `  G
) `  y ) ,  ( ( 1st `  F ) `  y
) >. (comp `  D
) ( ( 1st `  F ) `  z
) ) ( ( x  e.  B  |->  X ) `  y ) ) )
124123ralrimivvva 2801 . . 3  |-  ( ph  ->  A. y  e.  B  A. z  e.  B  A. f  e.  (
y (  Hom  `  C
) z ) ( ( ( x  e.  B  |->  X ) `  z ) ( <.
( ( 1st `  G
) `  y ) ,  ( ( 1st `  G ) `  z
) >. (comp `  D
) ( ( 1st `  F ) `  z
) ) ( ( y ( 2nd `  G
) z ) `  f ) )  =  ( ( ( y ( 2nd `  F
) z ) `  f ) ( <.
( ( 1st `  G
) `  y ) ,  ( ( 1st `  F ) `  y
) >. (comp `  D
) ( ( 1st `  F ) `  z
) ) ( ( x  e.  B  |->  X ) `  y ) ) )
12583, 10, 76, 21, 68, 16, 5isnat2 14147 . . 3  |-  ( ph  ->  ( ( x  e.  B  |->  X )  e.  ( G N F )  <->  ( ( x  e.  B  |->  X )  e.  X_ y  e.  B  ( ( ( 1st `  G ) `  y
) (  Hom  `  D
) ( ( 1st `  F ) `  y
) )  /\  A. y  e.  B  A. z  e.  B  A. f  e.  ( y
(  Hom  `  C ) z ) ( ( ( x  e.  B  |->  X ) `  z
) ( <. (
( 1st `  G
) `  y ) ,  ( ( 1st `  G ) `  z
) >. (comp `  D
) ( ( 1st `  F ) `  z
) ) ( ( y ( 2nd `  G
) z ) `  f ) )  =  ( ( ( y ( 2nd `  F
) z ) `  f ) ( <.
( ( 1st `  G
) `  y ) ,  ( ( 1st `  F ) `  y
) >. (comp `  D
) ( ( 1st `  F ) `  z
) ) ( ( x  e.  B  |->  X ) `  y ) ) ) ) )
12638, 124, 125mpbir2and 890 . 2  |-  ( ph  ->  ( x  e.  B  |->  X )  e.  ( G N F ) )
127 nfv 1630 . . . 4  |-  F/ y ( U `  x
) ( ( ( 1st `  F ) `
 x ) J ( ( 1st `  G
) `  x )
) ( ( x  e.  B  |->  X ) `
 x )
128127, 97, 101cbvral 2930 . . 3  |-  ( A. x  e.  B  ( U `  x )
( ( ( 1st `  F ) `  x
) J ( ( 1st `  G ) `
 x ) ) ( ( x  e.  B  |->  X ) `  x )  <->  A. y  e.  B  ( U `  y ) ( ( ( 1st `  F
) `  y ) J ( ( 1st `  G ) `  y
) ) ( ( x  e.  B  |->  X ) `  y ) )
12945, 128sylib 190 . 2  |-  ( ph  ->  A. y  e.  B  ( U `  y ) ( ( ( 1st `  F ) `  y
) J ( ( 1st `  G ) `
 y ) ) ( ( x  e.  B  |->  X ) `  y ) )
130 fuciso.q . . 3  |-  Q  =  ( C FuncCat  D )
131 fucinv.i . . 3  |-  I  =  (Inv `  Q )
132130, 10, 83, 5, 16, 131, 4fucinv 14172 . 2  |-  ( ph  ->  ( U ( F I G ) ( x  e.  B  |->  X )  <->  ( U  e.  ( F N G )  /\  ( x  e.  B  |->  X )  e.  ( G N F )  /\  A. y  e.  B  ( U `  y )
( ( ( 1st `  F ) `  y
) J ( ( 1st `  G ) `
 y ) ) ( ( x  e.  B  |->  X ) `  y ) ) ) )
1331, 126, 129, 132mpbir3and 1138 1  |-  ( ph  ->  U ( F I G ) ( x  e.  B  |->  X ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   A.wral 2707   _Vcvv 2958   <.cop 3819   class class class wbr 4214    e. cmpt 4268    X. cxp 4878   Rel wrel 4885   -->wf 5452   ` cfv 5456  (class class class)co 6083   1stc1st 6349   2ndc2nd 6350   X_cixp 7065   Basecbs 13471    Hom chom 13542  compcco 13543   Catccat 13891   Idccid 13892  Sectcsect 13972  Invcinv 13973    Func cfunc 14053   Nat cnat 14140   FuncCat cfuc 14141
This theorem is referenced by:  fuciso  14174  yonedainv  14380
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069
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 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-1o 6726  df-oadd 6730  df-er 6907  df-map 7022  df-ixp 7066  df-en 7112  df-dom 7113  df-sdom 7114  df-fin 7115  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-nn 10003  df-2 10060  df-3 10061  df-4 10062  df-5 10063  df-6 10064  df-7 10065  df-8 10066  df-9 10067  df-10 10068  df-n0 10224  df-z 10285  df-dec 10385  df-uz 10491  df-fz 11046  df-struct 13473  df-ndx 13474  df-slot 13475  df-base 13476  df-hom 13555  df-cco 13556  df-cat 13895  df-cid 13896  df-sect 13975  df-inv 13976  df-func 14057  df-nat 14142  df-fuc 14143
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