MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  invfuc Structured version   Unicode version

Theorem invfuc 15390
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 2457 . . . . . . . . . 10  |-  ( Base `  D )  =  (
Base `  D )
4 fucinv.j . . . . . . . . . 10  |-  J  =  (Inv `  D )
5 fuciso.f . . . . . . . . . . . . 13  |-  ( ph  ->  F  e.  ( C 
Func  D ) )
6 funcrcl 15279 . . . . . . . . . . . . 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 15278 . . . . . . . . . . . . 13  |-  Rel  ( C  Func  D )
12 1st2ndbr 6848 . . . . . . . . . . . . 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 15282 . . . . . . . . . . 11  |-  ( ph  ->  ( 1st `  F
) : B --> ( Base `  D ) )
1514ffvelrnda 6032 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  B )  ->  (
( 1st `  F
) `  x )  e.  ( Base `  D
) )
16 fuciso.g . . . . . . . . . . . . 13  |-  ( ph  ->  G  e.  ( C 
Func  D ) )
17 1st2ndbr 6848 . . . . . . . . . . . . 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 15282 . . . . . . . . . . 11  |-  ( ph  ->  ( 1st `  G
) : B --> ( Base `  D ) )
2019ffvelrnda 6032 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  B )  ->  (
( 1st `  G
) `  x )  e.  ( Base `  D
) )
21 eqid 2457 . . . . . . . . . 10  |-  ( Hom  `  D )  =  ( Hom  `  D )
223, 4, 9, 15, 20, 21invss 15177 . . . . . . . . 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 4497 . . . . . . . 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 5039 . . . . . . . 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 2871 . . . . 5  |-  ( ph  ->  A. x  e.  B  X  e.  ( (
( 1st `  G
) `  x )
( Hom  `  D ) ( ( 1st `  F
) `  x )
) )
29 fvex 5882 . . . . . . 7  |-  ( Base `  C )  e.  _V
3010, 29eqeltri 2541 . . . . . 6  |-  B  e. 
_V
31 mptelixpg 7525 . . . . . 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 5872 . . . . . 6  |-  ( x  =  y  ->  (
( 1st `  G
) `  x )  =  ( ( 1st `  G ) `  y
) )
35 fveq2 5872 . . . . . 6  |-  ( x  =  y  ->  (
( 1st `  F
) `  x )  =  ( ( 1st `  F ) `  y
) )
3634, 35oveq12d 6314 . . . . 5  |-  ( x  =  y  ->  (
( ( 1st `  G
) `  x )
( Hom  `  D ) ( ( 1st `  F
) `  x )
)  =  ( ( ( 1st `  G
) `  y )
( Hom  `  D ) ( ( 1st `  F
) `  y )
) )
3736cbvixpv 7506 . . . 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 2555 . . 3  |-  ( ph  ->  ( x  e.  B  |->  X )  e.  X_ y  e.  B  (
( ( 1st `  G
) `  y )
( Hom  `  D ) ( ( 1st `  F
) `  y )
) )
39 simpr2 1003 . . . . . . . . . . . . . 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 2457 . . . . . . . . . . . . . . . . . . 19  |-  ( x  e.  B  |->  X )  =  ( x  e.  B  |->  X )
4241fvmpt2 5964 . . . . . . . . . . . . . . . . . 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 4482 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  x  e.  B )  ->  ( U `  x )
( ( ( 1st `  F ) `  x
) J ( ( 1st `  G ) `
 x ) ) ( ( x  e.  B  |->  X ) `  x ) )
4544ralrimiva 2871 . . . . . . . . . . . . . . 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 2619 . . . . . . . . . . . . . . . 16  |-  F/_ x
( U `  z
)
48 nfcv 2619 . . . . . . . . . . . . . . . 16  |-  F/_ x
( ( ( 1st `  F ) `  z
) J ( ( 1st `  G ) `
 z ) )
49 nffvmpt1 5880 . . . . . . . . . . . . . . . 16  |-  F/_ x
( ( x  e.  B  |->  X ) `  z )
5047, 48, 49nfbr 4500 . . . . . . . . . . . . . . 15  |-  F/ x
( U `  z
) ( ( ( 1st `  F ) `
 z ) J ( ( 1st `  G
) `  z )
) ( ( x  e.  B  |->  X ) `
 z )
51 fveq2 5872 . . . . . . . . . . . . . . . 16  |-  ( x  =  z  ->  ( U `  x )  =  ( U `  z ) )
52 fveq2 5872 . . . . . . . . . . . . . . . . 17  |-  ( x  =  z  ->  (
( 1st `  F
) `  x )  =  ( ( 1st `  F ) `  z
) )
53 fveq2 5872 . . . . . . . . . . . . . . . . 17  |-  ( x  =  z  ->  (
( 1st `  G
) `  x )  =  ( ( 1st `  G ) `  z
) )
5452, 53oveq12d 6314 . . . . . . . . . . . . . . . 16  |-  ( x  =  z  ->  (
( ( 1st `  F
) `  x ) J ( ( 1st `  G ) `  x
) )  =  ( ( ( 1st `  F
) `  z ) J ( ( 1st `  G ) `  z
) ) )
55 fveq2 5872 . . . . . . . . . . . . . . . 16  |-  ( x  =  z  ->  (
( x  e.  B  |->  X ) `  x
)  =  ( ( x  e.  B  |->  X ) `  z ) )
5651, 54, 55breq123d 4470 . . . . . . . . . . . . . . 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 3204 . . . . . . . . . . . . . 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 6033 . . . . . . . . . . . . . 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 6033 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( y  e.  B  /\  z  e.  B  /\  f  e.  ( y ( Hom  `  C ) z ) ) )  ->  (
( 1st `  G
) `  z )  e.  ( Base `  D
) )
64 eqid 2457 . . . . . . . . . . . . . 14  |-  (Sect `  D )  =  (Sect `  D )
653, 4, 59, 61, 63, 64isinv 15176 . . . . . . . . . . . . 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 2457 . . . . . . . . . . . 12  |-  (comp `  D )  =  (comp `  D )
69 eqid 2457 . . . . . . . . . . . 12  |-  ( Id
`  D )  =  ( Id `  D
)
703, 21, 68, 69, 64, 59, 61, 63issect 15169 . . . . . . . . . . 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 1010 . . . . . . . . 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 6311 . . . . . . . 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 1002 . . . . . . . . . 10  |-  ( (
ph  /\  ( y  e.  B  /\  z  e.  B  /\  f  e.  ( y ( Hom  `  C ) z ) ) )  ->  y  e.  B )
7560, 74ffvelrnd 6033 . . . . . . . . 9  |-  ( (
ph  /\  ( y  e.  B  /\  z  e.  B  /\  f  e.  ( y ( Hom  `  C ) z ) ) )  ->  (
( 1st `  F
) `  y )  e.  ( Base `  D
) )
76 eqid 2457 . . . . . . . . . . 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 15284 . . . . . . . . . 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 1004 . . . . . . . . . 10  |-  ( (
ph  /\  ( y  e.  B  /\  z  e.  B  /\  f  e.  ( y ( Hom  `  C ) z ) ) )  ->  f  e.  ( y ( Hom  `  C ) z ) )
8078, 79ffvelrnd 6033 . . . . . . . . 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 15100 . . . . . . . 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 2499 . . . . . . 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 15367 . . . . . . . . 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 15369 . . . . . . . 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 1009 . . . . . . . 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 15103 . . . . . . 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 15371 . . . . . . . 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 6312 . . . . . . 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 2502 . . . . . 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 6311 . . . . 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 6033 . . . . . 6  |-  ( (
ph  /\  ( y  e.  B  /\  z  e.  B  /\  f  e.  ( y ( Hom  `  C ) z ) ) )  ->  (
( 1st `  G
) `  y )  e.  ( Base `  D
) )
94 nfcv 2619 . . . . . . . . . . . . 13  |-  F/_ x
( U `  y
)
95 nfcv 2619 . . . . . . . . . . . . 13  |-  F/_ x
( ( ( 1st `  F ) `  y
) J ( ( 1st `  G ) `
 y ) )
96 nffvmpt1 5880 . . . . . . . . . . . . 13  |-  F/_ x
( ( x  e.  B  |->  X ) `  y )
9794, 95, 96nfbr 4500 . . . . . . . . . . . 12  |-  F/ x
( U `  y
) ( ( ( 1st `  F ) `
 y ) J ( ( 1st `  G
) `  y )
) ( ( x  e.  B  |->  X ) `
 y )
98 fveq2 5872 . . . . . . . . . . . . 13  |-  ( x  =  y  ->  ( U `  x )  =  ( U `  y ) )
9935, 34oveq12d 6314 . . . . . . . . . . . . 13  |-  ( x  =  y  ->  (
( ( 1st `  F
) `  x ) J ( ( 1st `  G ) `  x
) )  =  ( ( ( 1st `  F
) `  y ) J ( ( 1st `  G ) `  y
) ) )
100 fveq2 5872 . . . . . . . . . . . . 13  |-  ( x  =  y  ->  (
( x  e.  B  |->  X ) `  x
)  =  ( ( x  e.  B  |->  X ) `  y ) )
10198, 99, 100breq123d 4470 . . . . . . . . . . . 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 3204 . . . . . . . . . . 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 15176 . . . . . . . . . 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 15169 . . . . . . . 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 1008 . . . . . 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 1009 . . . . . . 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 15284 . . . . . . . 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 6033 . . . . . . 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 15102 . . . . . 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 15103 . . . . 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 15369 . . . . . . . 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 15103 . . . . . . 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 1010 . . . . . . . 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 6312 . . . . . . 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 15101 . . . . . . 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 2502 . . . . . 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 6312 . . . . 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 2503 . . . 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 2879 . . 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 15364 . . 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 922 . 2  |-  ( ph  ->  ( x  e.  B  |->  X )  e.  ( G N F ) )
127 nfv 1708 . . . 4  |-  F/ y ( U `  x
) ( ( ( 1st `  F ) `
 x ) J ( ( 1st `  G
) `  x )
) ( ( x  e.  B  |->  X ) `
 x )
128127, 97, 101cbvral 3080 . . 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 15389 . 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 1179 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 973    = wceq 1395    e. wcel 1819   A.wral 2807   _Vcvv 3109   <.cop 4038   class class class wbr 4456    |-> cmpt 4515    X. cxp 5006   Rel wrel 5013   -->wf 5590   ` cfv 5594  (class class class)co 6296   1stc1st 6797   2ndc2nd 6798   X_cixp 7488   Basecbs 14644   Hom chom 14723  compcco 14724   Catccat 15081   Idccid 15082  Sectcsect 15160  Invcinv 15161    Func cfunc 15270   Nat cnat 15357   FuncCat cfuc 15358
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-fal 1401  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-map 7440  df-ixp 7489  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-3 10616  df-4 10617  df-5 10618  df-6 10619  df-7 10620  df-8 10621  df-9 10622  df-10 10623  df-n0 10817  df-z 10886  df-dec 11001  df-uz 11107  df-fz 11698  df-struct 14646  df-ndx 14647  df-slot 14648  df-base 14649  df-hom 14736  df-cco 14737  df-cat 15085  df-cid 15086  df-sect 15163  df-inv 15164  df-func 15274  df-nat 15359  df-fuc 15360
This theorem is referenced by:  fuciso  15391  yonedainv  15677
  Copyright terms: Public domain W3C validator