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

Theorem invfuc 14983
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 2451 . . . . . . . . . 10  |-  ( Base `  D )  =  (
Base `  D )
4 fucinv.j . . . . . . . . . 10  |-  J  =  (Inv `  D )
5 fuciso.f . . . . . . . . . . . . 13  |-  ( ph  ->  F  e.  ( C 
Func  D ) )
6 funcrcl 14872 . . . . . . . . . . . . 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 14871 . . . . . . . . . . . . 13  |-  Rel  ( C  Func  D )
12 1st2ndbr 6720 . . . . . . . . . . . . 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 14875 . . . . . . . . . . 11  |-  ( ph  ->  ( 1st `  F
) : B --> ( Base `  D ) )
1514ffvelrnda 5939 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  B )  ->  (
( 1st `  F
) `  x )  e.  ( Base `  D
) )
16 fuciso.g . . . . . . . . . . . . 13  |-  ( ph  ->  G  e.  ( C 
Func  D ) )
17 1st2ndbr 6720 . . . . . . . . . . . . 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 14875 . . . . . . . . . . 11  |-  ( ph  ->  ( 1st `  G
) : B --> ( Base `  D ) )
2019ffvelrnda 5939 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  B )  ->  (
( 1st `  G
) `  x )  e.  ( Base `  D
) )
21 eqid 2451 . . . . . . . . . 10  |-  ( Hom  `  D )  =  ( Hom  `  D )
223, 4, 9, 15, 20, 21invss 14798 . . . . . . . . 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 4428 . . . . . . . 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 4965 . . . . . . . 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 2820 . . . . 5  |-  ( ph  ->  A. x  e.  B  X  e.  ( (
( 1st `  G
) `  x )
( Hom  `  D ) ( ( 1st `  F
) `  x )
) )
29 fvex 5796 . . . . . . 7  |-  ( Base `  C )  e.  _V
3010, 29eqeltri 2533 . . . . . 6  |-  B  e. 
_V
31 mptelixpg 7397 . . . . . 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 5786 . . . . . 6  |-  ( x  =  y  ->  (
( 1st `  G
) `  x )  =  ( ( 1st `  G ) `  y
) )
35 fveq2 5786 . . . . . 6  |-  ( x  =  y  ->  (
( 1st `  F
) `  x )  =  ( ( 1st `  F ) `  y
) )
3634, 35oveq12d 6205 . . . . 5  |-  ( x  =  y  ->  (
( ( 1st `  G
) `  x )
( Hom  `  D ) ( ( 1st `  F
) `  x )
)  =  ( ( ( 1st `  G
) `  y )
( Hom  `  D ) ( ( 1st `  F
) `  y )
) )
3736cbvixpv 7378 . . . 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 2547 . . 3  |-  ( ph  ->  ( x  e.  B  |->  X )  e.  X_ y  e.  B  (
( ( 1st `  G
) `  y )
( Hom  `  D ) ( ( 1st `  F
) `  y )
) )
39 simpr2 995 . . . . . . . . . . . . . 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 2451 . . . . . . . . . . . . . . . . . . 19  |-  ( x  e.  B  |->  X )  =  ( x  e.  B  |->  X )
4241fvmpt2 5877 . . . . . . . . . . . . . . . . . 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 4413 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  x  e.  B )  ->  ( U `  x )
( ( ( 1st `  F ) `  x
) J ( ( 1st `  G ) `
 x ) ) ( ( x  e.  B  |->  X ) `  x ) )
4544ralrimiva 2820 . . . . . . . . . . . . . . 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 2611 . . . . . . . . . . . . . . . 16  |-  F/_ x
( U `  z
)
48 nfcv 2611 . . . . . . . . . . . . . . . 16  |-  F/_ x
( ( ( 1st `  F ) `  z
) J ( ( 1st `  G ) `
 z ) )
49 nffvmpt1 5794 . . . . . . . . . . . . . . . 16  |-  F/_ x
( ( x  e.  B  |->  X ) `  z )
5047, 48, 49nfbr 4431 . . . . . . . . . . . . . . 15  |-  F/ x
( U `  z
) ( ( ( 1st `  F ) `
 z ) J ( ( 1st `  G
) `  z )
) ( ( x  e.  B  |->  X ) `
 z )
51 fveq2 5786 . . . . . . . . . . . . . . . 16  |-  ( x  =  z  ->  ( U `  x )  =  ( U `  z ) )
52 fveq2 5786 . . . . . . . . . . . . . . . . 17  |-  ( x  =  z  ->  (
( 1st `  F
) `  x )  =  ( ( 1st `  F ) `  z
) )
53 fveq2 5786 . . . . . . . . . . . . . . . . 17  |-  ( x  =  z  ->  (
( 1st `  G
) `  x )  =  ( ( 1st `  G ) `  z
) )
5452, 53oveq12d 6205 . . . . . . . . . . . . . . . 16  |-  ( x  =  z  ->  (
( ( 1st `  F
) `  x ) J ( ( 1st `  G ) `  x
) )  =  ( ( ( 1st `  F
) `  z ) J ( ( 1st `  G ) `  z
) ) )
55 fveq2 5786 . . . . . . . . . . . . . . . 16  |-  ( x  =  z  ->  (
( x  e.  B  |->  X ) `  x
)  =  ( ( x  e.  B  |->  X ) `  z ) )
5651, 54, 55breq123d 4401 . . . . . . . . . . . . . . 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 3160 . . . . . . . . . . . . . 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 5940 . . . . . . . . . . . . . 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 5940 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( y  e.  B  /\  z  e.  B  /\  f  e.  ( y ( Hom  `  C ) z ) ) )  ->  (
( 1st `  G
) `  z )  e.  ( Base `  D
) )
64 eqid 2451 . . . . . . . . . . . . . 14  |-  (Sect `  D )  =  (Sect `  D )
653, 4, 59, 61, 63, 64isinv 14797 . . . . . . . . . . . . 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 2451 . . . . . . . . . . . 12  |-  (comp `  D )  =  (comp `  D )
69 eqid 2451 . . . . . . . . . . . 12  |-  ( Id
`  D )  =  ( Id `  D
)
703, 21, 68, 69, 64, 59, 61, 63issect 14791 . . . . . . . . . . 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 1002 . . . . . . . . 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 6202 . . . . . . . 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 994 . . . . . . . . . 10  |-  ( (
ph  /\  ( y  e.  B  /\  z  e.  B  /\  f  e.  ( y ( Hom  `  C ) z ) ) )  ->  y  e.  B )
7560, 74ffvelrnd 5940 . . . . . . . . 9  |-  ( (
ph  /\  ( y  e.  B  /\  z  e.  B  /\  f  e.  ( y ( Hom  `  C ) z ) ) )  ->  (
( 1st `  F
) `  y )  e.  ( Base `  D
) )
76 eqid 2451 . . . . . . . . . . 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 14877 . . . . . . . . . 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 996 . . . . . . . . . 10  |-  ( (
ph  /\  ( y  e.  B  /\  z  e.  B  /\  f  e.  ( y ( Hom  `  C ) z ) ) )  ->  f  e.  ( y ( Hom  `  C ) z ) )
8078, 79ffvelrnd 5940 . . . . . . . . 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 14720 . . . . . . . 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 2492 . . . . . . 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 14960 . . . . . . . . 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 14962 . . . . . . . 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 1001 . . . . . . . 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 14723 . . . . . . 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 14964 . . . . . . . 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 6203 . . . . . . 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 2495 . . . . . 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 6202 . . . . 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 5940 . . . . . 6  |-  ( (
ph  /\  ( y  e.  B  /\  z  e.  B  /\  f  e.  ( y ( Hom  `  C ) z ) ) )  ->  (
( 1st `  G
) `  y )  e.  ( Base `  D
) )
94 nfcv 2611 . . . . . . . . . . . . 13  |-  F/_ x
( U `  y
)
95 nfcv 2611 . . . . . . . . . . . . 13  |-  F/_ x
( ( ( 1st `  F ) `  y
) J ( ( 1st `  G ) `
 y ) )
96 nffvmpt1 5794 . . . . . . . . . . . . 13  |-  F/_ x
( ( x  e.  B  |->  X ) `  y )
9794, 95, 96nfbr 4431 . . . . . . . . . . . 12  |-  F/ x
( U `  y
) ( ( ( 1st `  F ) `
 y ) J ( ( 1st `  G
) `  y )
) ( ( x  e.  B  |->  X ) `
 y )
98 fveq2 5786 . . . . . . . . . . . . 13  |-  ( x  =  y  ->  ( U `  x )  =  ( U `  y ) )
9935, 34oveq12d 6205 . . . . . . . . . . . . 13  |-  ( x  =  y  ->  (
( ( 1st `  F
) `  x ) J ( ( 1st `  G ) `  x
) )  =  ( ( ( 1st `  F
) `  y ) J ( ( 1st `  G ) `  y
) ) )
100 fveq2 5786 . . . . . . . . . . . . 13  |-  ( x  =  y  ->  (
( x  e.  B  |->  X ) `  x
)  =  ( ( x  e.  B  |->  X ) `  y ) )
10198, 99, 100breq123d 4401 . . . . . . . . . . . 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 3160 . . . . . . . . . . 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 14797 . . . . . . . . . 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 14791 . . . . . . . 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 1000 . . . . . 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 1001 . . . . . . 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 14877 . . . . . . . 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 5940 . . . . . . 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 14722 . . . . . 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 14723 . . . . 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 14962 . . . . . . . 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 14723 . . . . . . 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 1002 . . . . . . . 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 6203 . . . . . . 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 14721 . . . . . . 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 2495 . . . . . 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 6203 . . . . 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 2496 . . . 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 2902 . . 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 14957 . . 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 913 . 2  |-  ( ph  ->  ( x  e.  B  |->  X )  e.  ( G N F ) )
127 nfv 1674 . . . 4  |-  F/ y ( U `  x
) ( ( ( 1st `  F ) `
 x ) J ( ( 1st `  G
) `  x )
) ( ( x  e.  B  |->  X ) `
 x )
128127, 97, 101cbvral 3036 . . 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 14982 . 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 1171 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 965    = wceq 1370    e. wcel 1758   A.wral 2793   _Vcvv 3065   <.cop 3978   class class class wbr 4387    |-> cmpt 4445    X. cxp 4933   Rel wrel 4940   -->wf 5509   ` cfv 5513  (class class class)co 6187   1stc1st 6672   2ndc2nd 6673   X_cixp 7360   Basecbs 14273   Hom chom 14348  compcco 14349   Catccat 14701   Idccid 14702  Sectcsect 14782  Invcinv 14783    Func cfunc 14863   Nat cnat 14950   FuncCat cfuc 14951
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4498  ax-sep 4508  ax-nul 4516  ax-pow 4565  ax-pr 4626  ax-un 6469  ax-cnex 9436  ax-resscn 9437  ax-1cn 9438  ax-icn 9439  ax-addcl 9440  ax-addrcl 9441  ax-mulcl 9442  ax-mulrcl 9443  ax-mulcom 9444  ax-addass 9445  ax-mulass 9446  ax-distr 9447  ax-i2m1 9448  ax-1ne0 9449  ax-1rid 9450  ax-rnegex 9451  ax-rrecex 9452  ax-cnre 9453  ax-pre-lttri 9454  ax-pre-lttrn 9455  ax-pre-ltadd 9456  ax-pre-mulgt0 9457
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2599  df-ne 2644  df-nel 2645  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3067  df-sbc 3282  df-csb 3384  df-dif 3426  df-un 3428  df-in 3430  df-ss 3437  df-pss 3439  df-nul 3733  df-if 3887  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4187  df-int 4224  df-iun 4268  df-br 4388  df-opab 4446  df-mpt 4447  df-tr 4481  df-eprel 4727  df-id 4731  df-po 4736  df-so 4737  df-fr 4774  df-we 4776  df-ord 4817  df-on 4818  df-lim 4819  df-suc 4820  df-xp 4941  df-rel 4942  df-cnv 4943  df-co 4944  df-dm 4945  df-rn 4946  df-res 4947  df-ima 4948  df-iota 5476  df-fun 5515  df-fn 5516  df-f 5517  df-f1 5518  df-fo 5519  df-f1o 5520  df-fv 5521  df-riota 6148  df-ov 6190  df-oprab 6191  df-mpt2 6192  df-om 6574  df-1st 6674  df-2nd 6675  df-recs 6929  df-rdg 6963  df-1o 7017  df-oadd 7021  df-er 7198  df-map 7313  df-ixp 7361  df-en 7408  df-dom 7409  df-sdom 7410  df-fin 7411  df-pnf 9518  df-mnf 9519  df-xr 9520  df-ltxr 9521  df-le 9522  df-sub 9695  df-neg 9696  df-nn 10421  df-2 10478  df-3 10479  df-4 10480  df-5 10481  df-6 10482  df-7 10483  df-8 10484  df-9 10485  df-10 10486  df-n0 10678  df-z 10745  df-dec 10854  df-uz 10960  df-fz 11536  df-struct 14275  df-ndx 14276  df-slot 14277  df-base 14278  df-hom 14361  df-cco 14362  df-cat 14705  df-cid 14706  df-sect 14785  df-inv 14786  df-func 14867  df-nat 14952  df-fuc 14953
This theorem is referenced by:  fuciso  14984  yonedainv  15190
  Copyright terms: Public domain W3C validator