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Theorem invfuc 15212
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 2441 . . . . . . . . . 10  |-  ( Base `  D )  =  (
Base `  D )
4 fucinv.j . . . . . . . . . 10  |-  J  =  (Inv `  D )
5 fuciso.f . . . . . . . . . . . . 13  |-  ( ph  ->  F  e.  ( C 
Func  D ) )
6 funcrcl 15101 . . . . . . . . . . . . 13  |-  ( F  e.  ( C  Func  D )  ->  ( C  e.  Cat  /\  D  e. 
Cat ) )
75, 6syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  ( C  e.  Cat  /\  D  e.  Cat )
)
87simprd 463 . . . . . . . . . . 11  |-  ( ph  ->  D  e.  Cat )
98adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  B )  ->  D  e.  Cat )
10 fuciso.b . . . . . . . . . . . 12  |-  B  =  ( Base `  C
)
11 relfunc 15100 . . . . . . . . . . . . 13  |-  Rel  ( C  Func  D )
12 1st2ndbr 6830 . . . . . . . . . . . . 13  |-  ( ( Rel  ( C  Func  D )  /\  F  e.  ( C  Func  D
) )  ->  ( 1st `  F ) ( C  Func  D )
( 2nd `  F
) )
1311, 5, 12sylancr 663 . . . . . . . . . . . 12  |-  ( ph  ->  ( 1st `  F
) ( C  Func  D ) ( 2nd `  F
) )
1410, 3, 13funcf1 15104 . . . . . . . . . . 11  |-  ( ph  ->  ( 1st `  F
) : B --> ( Base `  D ) )
1514ffvelrnda 6012 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  B )  ->  (
( 1st `  F
) `  x )  e.  ( Base `  D
) )
16 fuciso.g . . . . . . . . . . . . 13  |-  ( ph  ->  G  e.  ( C 
Func  D ) )
17 1st2ndbr 6830 . . . . . . . . . . . . 13  |-  ( ( Rel  ( C  Func  D )  /\  G  e.  ( C  Func  D
) )  ->  ( 1st `  G ) ( C  Func  D )
( 2nd `  G
) )
1811, 16, 17sylancr 663 . . . . . . . . . . . 12  |-  ( ph  ->  ( 1st `  G
) ( C  Func  D ) ( 2nd `  G
) )
1910, 3, 18funcf1 15104 . . . . . . . . . . 11  |-  ( ph  ->  ( 1st `  G
) : B --> ( Base `  D ) )
2019ffvelrnda 6012 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  B )  ->  (
( 1st `  G
) `  x )  e.  ( Base `  D
) )
21 eqid 2441 . . . . . . . . . 10  |-  ( Hom  `  D )  =  ( Hom  `  D )
223, 4, 9, 15, 20, 21invss 15027 . . . . . . . . 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 4474 . . . . . . . 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 5016 . . . . . . . 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 464 . . . . . . 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 2855 . . . . 5  |-  ( ph  ->  A. x  e.  B  X  e.  ( (
( 1st `  G
) `  x )
( Hom  `  D ) ( ( 1st `  F
) `  x )
) )
29 fvex 5862 . . . . . . 7  |-  ( Base `  C )  e.  _V
3010, 29eqeltri 2525 . . . . . 6  |-  B  e. 
_V
31 mptelixpg 7504 . . . . . 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 5 . . . . 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 212 . . . 4  |-  ( ph  ->  ( x  e.  B  |->  X )  e.  X_ x  e.  B  (
( ( 1st `  G
) `  x )
( Hom  `  D ) ( ( 1st `  F
) `  x )
) )
34 fveq2 5852 . . . . . 6  |-  ( x  =  y  ->  (
( 1st `  G
) `  x )  =  ( ( 1st `  G ) `  y
) )
35 fveq2 5852 . . . . . 6  |-  ( x  =  y  ->  (
( 1st `  F
) `  x )  =  ( ( 1st `  F ) `  y
) )
3634, 35oveq12d 6295 . . . . 5  |-  ( x  =  y  ->  (
( ( 1st `  G
) `  x )
( Hom  `  D ) ( ( 1st `  F
) `  x )
)  =  ( ( ( 1st `  G
) `  y )
( Hom  `  D ) ( ( 1st `  F
) `  y )
) )
3736cbvixpv 7485 . . . 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 2539 . . 3  |-  ( ph  ->  ( x  e.  B  |->  X )  e.  X_ y  e.  B  (
( ( 1st `  G
) `  y )
( Hom  `  D ) ( ( 1st `  F
) `  y )
) )
39 simpr2 1002 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( y  e.  B  /\  z  e.  B  /\  f  e.  ( y ( Hom  `  C ) z ) ) )  ->  z  e.  B )
40 simpr 461 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  x  e.  B )  ->  x  e.  B )
41 eqid 2441 . . . . . . . . . . . . . . . . . . 19  |-  ( x  e.  B  |->  X )  =  ( x  e.  B  |->  X )
4241fvmpt2 5944 . . . . . . . . . . . . . . . . . 18  |-  ( ( x  e.  B  /\  X  e.  ( (
( 1st `  G
) `  x )
( Hom  `  D ) ( ( 1st `  F
) `  x )
) )  ->  (
( x  e.  B  |->  X ) `  x
)  =  X )
4340, 27, 42syl2anc 661 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  x  e.  B )  ->  (
( x  e.  B  |->  X ) `  x
)  =  X )
442, 43breqtrrd 4459 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  x  e.  B )  ->  ( U `  x )
( ( ( 1st `  F ) `  x
) J ( ( 1st `  G ) `
 x ) ) ( ( x  e.  B  |->  X ) `  x ) )
4544ralrimiva 2855 . . . . . . . . . . . . . . 15  |-  ( ph  ->  A. x  e.  B  ( U `  x ) ( ( ( 1st `  F ) `  x
) J ( ( 1st `  G ) `
 x ) ) ( ( x  e.  B  |->  X ) `  x ) )
4645adantr 465 . . . . . . . . . . . . . 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 2603 . . . . . . . . . . . . . . . 16  |-  F/_ x
( U `  z
)
48 nfcv 2603 . . . . . . . . . . . . . . . 16  |-  F/_ x
( ( ( 1st `  F ) `  z
) J ( ( 1st `  G ) `
 z ) )
49 nffvmpt1 5860 . . . . . . . . . . . . . . . 16  |-  F/_ x
( ( x  e.  B  |->  X ) `  z )
5047, 48, 49nfbr 4477 . . . . . . . . . . . . . . 15  |-  F/ x
( U `  z
) ( ( ( 1st `  F ) `
 z ) J ( ( 1st `  G
) `  z )
) ( ( x  e.  B  |->  X ) `
 z )
51 fveq2 5852 . . . . . . . . . . . . . . . 16  |-  ( x  =  z  ->  ( U `  x )  =  ( U `  z ) )
52 fveq2 5852 . . . . . . . . . . . . . . . . 17  |-  ( x  =  z  ->  (
( 1st `  F
) `  x )  =  ( ( 1st `  F ) `  z
) )
53 fveq2 5852 . . . . . . . . . . . . . . . . 17  |-  ( x  =  z  ->  (
( 1st `  G
) `  x )  =  ( ( 1st `  G ) `  z
) )
5452, 53oveq12d 6295 . . . . . . . . . . . . . . . 16  |-  ( x  =  z  ->  (
( ( 1st `  F
) `  x ) J ( ( 1st `  G ) `  x
) )  =  ( ( ( 1st `  F
) `  z ) J ( ( 1st `  G ) `  z
) ) )
55 fveq2 5852 . . . . . . . . . . . . . . . 16  |-  ( x  =  z  ->  (
( x  e.  B  |->  X ) `  x
)  =  ( ( x  e.  B  |->  X ) `  z ) )
5651, 54, 55breq123d 4447 . . . . . . . . . . . . . . 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 3188 . . . . . . . . . . . . . 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 60 . . . . . . . . . . . . 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 465 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( y  e.  B  /\  z  e.  B  /\  f  e.  ( y ( Hom  `  C ) z ) ) )  ->  D  e.  Cat )
6014adantr 465 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( y  e.  B  /\  z  e.  B  /\  f  e.  ( y ( Hom  `  C ) z ) ) )  ->  ( 1st `  F ) : B --> ( Base `  D
) )
6160, 39ffvelrnd 6013 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( y  e.  B  /\  z  e.  B  /\  f  e.  ( y ( Hom  `  C ) z ) ) )  ->  (
( 1st `  F
) `  z )  e.  ( Base `  D
) )
6219adantr 465 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( y  e.  B  /\  z  e.  B  /\  f  e.  ( y ( Hom  `  C ) z ) ) )  ->  ( 1st `  G ) : B --> ( Base `  D
) )
6362, 39ffvelrnd 6013 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( y  e.  B  /\  z  e.  B  /\  f  e.  ( y ( Hom  `  C ) z ) ) )  ->  (
( 1st `  G
) `  z )  e.  ( Base `  D
) )
64 eqid 2441 . . . . . . . . . . . . . 14  |-  (Sect `  D )  =  (Sect `  D )
653, 4, 59, 61, 63, 64isinv 15026 . . . . . . . . . . . . 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 210 . . . . . . . . . . . 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 459 . . . . . . . . . . 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 2441 . . . . . . . . . . . 12  |-  (comp `  D )  =  (comp `  D )
69 eqid 2441 . . . . . . . . . . . 12  |-  ( Id
`  D )  =  ( Id `  D
)
703, 21, 68, 69, 64, 59, 61, 63issect 15020 . . . . . . . . . . 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 210 . . . . . . . . . 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 1009 . . . . . . . . 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 6292 . . . . . . . 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 1001 . . . . . . . . . 10  |-  ( (
ph  /\  ( y  e.  B  /\  z  e.  B  /\  f  e.  ( y ( Hom  `  C ) z ) ) )  ->  y  e.  B )
7560, 74ffvelrnd 6013 . . . . . . . . 9  |-  ( (
ph  /\  ( y  e.  B  /\  z  e.  B  /\  f  e.  ( y ( Hom  `  C ) z ) ) )  ->  (
( 1st `  F
) `  y )  e.  ( Base `  D
) )
76 eqid 2441 . . . . . . . . . . 11  |-  ( Hom  `  C )  =  ( Hom  `  C )
7713adantr 465 . . . . . . . . . . 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 15106 . . . . . . . . . 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 1003 . . . . . . . . . 10  |-  ( (
ph  /\  ( y  e.  B  /\  z  e.  B  /\  f  e.  ( y ( Hom  `  C ) z ) ) )  ->  f  e.  ( y ( Hom  `  C ) z ) )
8078, 79ffvelrnd 6013 . . . . . . . . 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 14952 . . . . . . . 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 2483 . . . . . . 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 465 . . . . . . . . . 10  |-  ( (
ph  /\  ( y  e.  B  /\  z  e.  B  /\  f  e.  ( y ( Hom  `  C ) z ) ) )  ->  U  e.  ( F N G ) )
8583, 84nat1st2nd 15189 . . . . . . . . 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 15191 . . . . . . . 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 1008 . . . . . . . 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 14955 . . . . . . 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 15193 . . . . . . . 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 6293 . . . . . . 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 2486 . . . . . 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 6292 . . . . 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 6013 . . . . . 6  |-  ( (
ph  /\  ( y  e.  B  /\  z  e.  B  /\  f  e.  ( y ( Hom  `  C ) z ) ) )  ->  (
( 1st `  G
) `  y )  e.  ( Base `  D
) )
94 nfcv 2603 . . . . . . . . . . . . 13  |-  F/_ x
( U `  y
)
95 nfcv 2603 . . . . . . . . . . . . 13  |-  F/_ x
( ( ( 1st `  F ) `  y
) J ( ( 1st `  G ) `
 y ) )
96 nffvmpt1 5860 . . . . . . . . . . . . 13  |-  F/_ x
( ( x  e.  B  |->  X ) `  y )
9794, 95, 96nfbr 4477 . . . . . . . . . . . 12  |-  F/ x
( U `  y
) ( ( ( 1st `  F ) `
 y ) J ( ( 1st `  G
) `  y )
) ( ( x  e.  B  |->  X ) `
 y )
98 fveq2 5852 . . . . . . . . . . . . 13  |-  ( x  =  y  ->  ( U `  x )  =  ( U `  y ) )
9935, 34oveq12d 6295 . . . . . . . . . . . . 13  |-  ( x  =  y  ->  (
( ( 1st `  F
) `  x ) J ( ( 1st `  G ) `  x
) )  =  ( ( ( 1st `  F
) `  y ) J ( ( 1st `  G ) `  y
) ) )
100 fveq2 5852 . . . . . . . . . . . . 13  |-  ( x  =  y  ->  (
( x  e.  B  |->  X ) `  x
)  =  ( ( x  e.  B  |->  X ) `  y ) )
10198, 99, 100breq123d 4447 . . . . . . . . . . . 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 3188 . . . . . . . . . . 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 60 . . . . . . . . . 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 15026 . . . . . . . . . 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 210 . . . . . . . . 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 463 . . . . . . . 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 15020 . . . . . . . 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 210 . . . . . . 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 1007 . . . . . 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 1008 . . . . . . 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 465 . . . . . . . . 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 15106 . . . . . . . 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 6013 . . . . . . 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 14954 . . . . . 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 14955 . . . . 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 15191 . . . . . . . 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 14955 . . . . . . 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 1009 . . . . . . . 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 6293 . . . . . . 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 14953 . . . . . . 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 2486 . . . . . 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 6293 . . . . 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 2487 . . . 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 2863 . . 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 15186 . . 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 920 . 2  |-  ( ph  ->  ( x  e.  B  |->  X )  e.  ( G N F ) )
127 nfv 1692 . . . 4  |-  F/ y ( U `  x
) ( ( ( 1st `  F ) `
 x ) J ( ( 1st `  G
) `  x )
) ( ( x  e.  B  |->  X ) `
 x )
128127, 97, 101cbvral 3064 . . 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 196 . 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 15211 . 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 1178 1  |-  ( ph  ->  U ( F I G ) ( x  e.  B  |->  X ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 972    = wceq 1381    e. wcel 1802   A.wral 2791   _Vcvv 3093   <.cop 4016   class class class wbr 4433    |-> cmpt 4491    X. cxp 4983   Rel wrel 4990   -->wf 5570   ` cfv 5574  (class class class)co 6277   1stc1st 6779   2ndc2nd 6780   X_cixp 7467   Basecbs 14504   Hom chom 14580  compcco 14581   Catccat 14933   Idccid 14934  Sectcsect 15011  Invcinv 15012    Func cfunc 15092   Nat cnat 15179   FuncCat cfuc 15180
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-fal 1387  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-pss 3474  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-tp 4015  df-op 4017  df-uni 4231  df-int 4268  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-tr 4527  df-eprel 4777  df-id 4781  df-po 4786  df-so 4787  df-fr 4824  df-we 4826  df-ord 4867  df-on 4868  df-lim 4869  df-suc 4870  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6682  df-1st 6781  df-2nd 6782  df-recs 7040  df-rdg 7074  df-1o 7128  df-oadd 7132  df-er 7309  df-map 7420  df-ixp 7468  df-en 7515  df-dom 7516  df-sdom 7517  df-fin 7518  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9807  df-neg 9808  df-nn 10538  df-2 10595  df-3 10596  df-4 10597  df-5 10598  df-6 10599  df-7 10600  df-8 10601  df-9 10602  df-10 10603  df-n0 10797  df-z 10866  df-dec 10980  df-uz 11086  df-fz 11677  df-struct 14506  df-ndx 14507  df-slot 14508  df-base 14509  df-hom 14593  df-cco 14594  df-cat 14937  df-cid 14938  df-sect 15014  df-inv 15015  df-func 15096  df-nat 15181  df-fuc 15182
This theorem is referenced by:  fuciso  15213  yonedainv  15419
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