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Theorem fucpropd 13867
Description: If two categories have the same set of objects, morphisms, and compositions, then they have the same functor categories. (Contributed by Mario Carneiro, 26-Jan-2017.)
Hypotheses
Ref Expression
fucpropd.1  |-  ( ph  ->  (  Homf 
`  A )  =  (  Homf 
`  B ) )
fucpropd.2  |-  ( ph  ->  (compf `  A )  =  (compf `  B ) )
fucpropd.3  |-  ( ph  ->  (  Homf 
`  C )  =  (  Homf 
`  D ) )
fucpropd.4  |-  ( ph  ->  (compf `  C )  =  (compf `  D ) )
fucpropd.a  |-  ( ph  ->  A  e.  Cat )
fucpropd.b  |-  ( ph  ->  B  e.  Cat )
fucpropd.c  |-  ( ph  ->  C  e.  Cat )
fucpropd.d  |-  ( ph  ->  D  e.  Cat )
Assertion
Ref Expression
fucpropd  |-  ( ph  ->  ( A FuncCat  C )  =  ( B FuncCat  D
) )

Proof of Theorem fucpropd
Dummy variables  a 
b  f  g  h  v  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fucpropd.1 . . . . 5  |-  ( ph  ->  (  Homf 
`  A )  =  (  Homf 
`  B ) )
2 fucpropd.2 . . . . 5  |-  ( ph  ->  (compf `  A )  =  (compf `  B ) )
3 fucpropd.3 . . . . 5  |-  ( ph  ->  (  Homf 
`  C )  =  (  Homf 
`  D ) )
4 fucpropd.4 . . . . 5  |-  ( ph  ->  (compf `  C )  =  (compf `  D ) )
5 fucpropd.a . . . . 5  |-  ( ph  ->  A  e.  Cat )
6 fucpropd.b . . . . 5  |-  ( ph  ->  B  e.  Cat )
7 fucpropd.c . . . . 5  |-  ( ph  ->  C  e.  Cat )
8 fucpropd.d . . . . 5  |-  ( ph  ->  D  e.  Cat )
91, 2, 3, 4, 5, 6, 7, 8funcpropd 13790 . . . 4  |-  ( ph  ->  ( A  Func  C
)  =  ( B 
Func  D ) )
109opeq2d 3819 . . 3  |-  ( ph  -> 
<. ( Base `  ndx ) ,  ( A  Func  C ) >.  =  <. (
Base `  ndx ) ,  ( B  Func  D
) >. )
111, 2, 3, 4, 5, 6, 7, 8natpropd 13866 . . . 4  |-  ( ph  ->  ( A Nat  C )  =  ( B Nat  D
) )
1211opeq2d 3819 . . 3  |-  ( ph  -> 
<. (  Hom  `  ndx ) ,  ( A Nat  C ) >.  =  <. (  Hom  `  ndx ) ,  ( B Nat  D )
>. )
139, 9xpeq12d 4730 . . . . 5  |-  ( ph  ->  ( ( A  Func  C )  X.  ( A 
Func  C ) )  =  ( ( B  Func  D )  X.  ( B 
Func  D ) ) )
149adantr 451 . . . . 5  |-  ( (
ph  /\  v  e.  ( ( A  Func  C )  X.  ( A 
Func  C ) ) )  ->  ( A  Func  C )  =  ( B 
Func  D ) )
15 nfv 1609 . . . . . 6  |-  F/ f ( ph  /\  (
v  e.  ( ( A  Func  C )  X.  ( A  Func  C
) )  /\  h  e.  ( A  Func  C
) ) )
16 nfcsb1v 3126 . . . . . . 7  |-  F/_ f [_ ( 1st `  v
)  /  f ]_ [_ ( 2nd `  v
)  /  g ]_ ( b  e.  ( g ( B Nat  D
) h ) ,  a  e.  ( f ( B Nat  D ) g )  |->  ( x  e.  ( Base `  B
)  |->  ( ( b `
 x ) (
<. ( ( 1st `  f
) `  x ) ,  ( ( 1st `  g ) `  x
) >. (comp `  D
) ( ( 1st `  h ) `  x
) ) ( a `
 x ) ) ) )
1716a1i 10 . . . . . 6  |-  ( (
ph  /\  ( v  e.  ( ( A  Func  C )  X.  ( A 
Func  C ) )  /\  h  e.  ( A  Func  C ) ) )  ->  F/_ f [_ ( 1st `  v )  / 
f ]_ [_ ( 2nd `  v )  /  g ]_ ( b  e.  ( g ( B Nat  D
) h ) ,  a  e.  ( f ( B Nat  D ) g )  |->  ( x  e.  ( Base `  B
)  |->  ( ( b `
 x ) (
<. ( ( 1st `  f
) `  x ) ,  ( ( 1st `  g ) `  x
) >. (comp `  D
) ( ( 1st `  h ) `  x
) ) ( a `
 x ) ) ) ) )
18 fvex 5555 . . . . . . 7  |-  ( 1st `  v )  e.  _V
1918a1i 10 . . . . . 6  |-  ( (
ph  /\  ( v  e.  ( ( A  Func  C )  X.  ( A 
Func  C ) )  /\  h  e.  ( A  Func  C ) ) )  ->  ( 1st `  v
)  e.  _V )
20 nfv 1609 . . . . . . . 8  |-  F/ g ( ( ph  /\  ( v  e.  ( ( A  Func  C
)  X.  ( A 
Func  C ) )  /\  h  e.  ( A  Func  C ) ) )  /\  f  =  ( 1st `  v ) )
21 nfcsb1v 3126 . . . . . . . . 9  |-  F/_ g [_ ( 2nd `  v
)  /  g ]_ ( b  e.  ( g ( B Nat  D
) h ) ,  a  e.  ( f ( B Nat  D ) g )  |->  ( x  e.  ( Base `  B
)  |->  ( ( b `
 x ) (
<. ( ( 1st `  f
) `  x ) ,  ( ( 1st `  g ) `  x
) >. (comp `  D
) ( ( 1st `  h ) `  x
) ) ( a `
 x ) ) ) )
2221a1i 10 . . . . . . . 8  |-  ( ( ( ph  /\  (
v  e.  ( ( A  Func  C )  X.  ( A  Func  C
) )  /\  h  e.  ( A  Func  C
) ) )  /\  f  =  ( 1st `  v ) )  ->  F/_ g [_ ( 2nd `  v )  /  g ]_ ( b  e.  ( g ( B Nat  D
) h ) ,  a  e.  ( f ( B Nat  D ) g )  |->  ( x  e.  ( Base `  B
)  |->  ( ( b `
 x ) (
<. ( ( 1st `  f
) `  x ) ,  ( ( 1st `  g ) `  x
) >. (comp `  D
) ( ( 1st `  h ) `  x
) ) ( a `
 x ) ) ) ) )
23 fvex 5555 . . . . . . . . 9  |-  ( 2nd `  v )  e.  _V
2423a1i 10 . . . . . . . 8  |-  ( ( ( ph  /\  (
v  e.  ( ( A  Func  C )  X.  ( A  Func  C
) )  /\  h  e.  ( A  Func  C
) ) )  /\  f  =  ( 1st `  v ) )  -> 
( 2nd `  v
)  e.  _V )
2511ad3antrrr 710 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  ( v  e.  ( ( A  Func  C
)  X.  ( A 
Func  C ) )  /\  h  e.  ( A  Func  C ) ) )  /\  f  =  ( 1st `  v ) )  /\  g  =  ( 2nd `  v
) )  ->  ( A Nat  C )  =  ( B Nat  D ) )
2625oveqd 5891 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  ( v  e.  ( ( A  Func  C
)  X.  ( A 
Func  C ) )  /\  h  e.  ( A  Func  C ) ) )  /\  f  =  ( 1st `  v ) )  /\  g  =  ( 2nd `  v
) )  ->  (
g ( A Nat  C
) h )  =  ( g ( B Nat 
D ) h ) )
2725adantr 451 . . . . . . . . . . 11  |-  ( ( ( ( ( ph  /\  ( v  e.  ( ( A  Func  C
)  X.  ( A 
Func  C ) )  /\  h  e.  ( A  Func  C ) ) )  /\  f  =  ( 1st `  v ) )  /\  g  =  ( 2nd `  v
) )  /\  b  e.  ( g ( A Nat 
C ) h ) )  ->  ( A Nat  C )  =  ( B Nat 
D ) )
2827oveqd 5891 . . . . . . . . . 10  |-  ( ( ( ( ( ph  /\  ( v  e.  ( ( A  Func  C
)  X.  ( A 
Func  C ) )  /\  h  e.  ( A  Func  C ) ) )  /\  f  =  ( 1st `  v ) )  /\  g  =  ( 2nd `  v
) )  /\  b  e.  ( g ( A Nat 
C ) h ) )  ->  ( f
( A Nat  C ) g )  =  ( f ( B Nat  D
) g ) )
291homfeqbas 13615 . . . . . . . . . . . 12  |-  ( ph  ->  ( Base `  A
)  =  ( Base `  B ) )
3029ad4antr 712 . . . . . . . . . . 11  |-  ( ( ( ( ( ph  /\  ( v  e.  ( ( A  Func  C
)  X.  ( A 
Func  C ) )  /\  h  e.  ( A  Func  C ) ) )  /\  f  =  ( 1st `  v ) )  /\  g  =  ( 2nd `  v
) )  /\  (
b  e.  ( g ( A Nat  C ) h )  /\  a  e.  ( f ( A Nat 
C ) g ) ) )  ->  ( Base `  A )  =  ( Base `  B
) )
31 eqid 2296 . . . . . . . . . . . 12  |-  ( Base `  C )  =  (
Base `  C )
32 eqid 2296 . . . . . . . . . . . 12  |-  (  Hom  `  C )  =  (  Hom  `  C )
33 eqid 2296 . . . . . . . . . . . 12  |-  (comp `  C )  =  (comp `  C )
34 eqid 2296 . . . . . . . . . . . 12  |-  (comp `  D )  =  (comp `  D )
353ad5antr 714 . . . . . . . . . . . 12  |-  ( ( ( ( ( (
ph  /\  ( v  e.  ( ( A  Func  C )  X.  ( A 
Func  C ) )  /\  h  e.  ( A  Func  C ) ) )  /\  f  =  ( 1st `  v ) )  /\  g  =  ( 2nd `  v
) )  /\  (
b  e.  ( g ( A Nat  C ) h )  /\  a  e.  ( f ( A Nat 
C ) g ) ) )  /\  x  e.  ( Base `  A
) )  ->  (  Homf  `  C )  =  (  Homf 
`  D ) )
364ad5antr 714 . . . . . . . . . . . 12  |-  ( ( ( ( ( (
ph  /\  ( v  e.  ( ( A  Func  C )  X.  ( A 
Func  C ) )  /\  h  e.  ( A  Func  C ) ) )  /\  f  =  ( 1st `  v ) )  /\  g  =  ( 2nd `  v
) )  /\  (
b  e.  ( g ( A Nat  C ) h )  /\  a  e.  ( f ( A Nat 
C ) g ) ) )  /\  x  e.  ( Base `  A
) )  ->  (compf `  C
)  =  (compf `  D
) )
37 eqid 2296 . . . . . . . . . . . . . 14  |-  ( Base `  A )  =  (
Base `  A )
38 relfunc 13752 . . . . . . . . . . . . . . 15  |-  Rel  ( A  Func  C )
39 simpllr 735 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( ph  /\  ( v  e.  ( ( A  Func  C
)  X.  ( A 
Func  C ) )  /\  h  e.  ( A  Func  C ) ) )  /\  f  =  ( 1st `  v ) )  /\  g  =  ( 2nd `  v
) )  /\  (
b  e.  ( g ( A Nat  C ) h )  /\  a  e.  ( f ( A Nat 
C ) g ) ) )  ->  f  =  ( 1st `  v
) )
40 simp-4r 743 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ( ph  /\  ( v  e.  ( ( A  Func  C
)  X.  ( A 
Func  C ) )  /\  h  e.  ( A  Func  C ) ) )  /\  f  =  ( 1st `  v ) )  /\  g  =  ( 2nd `  v
) )  /\  (
b  e.  ( g ( A Nat  C ) h )  /\  a  e.  ( f ( A Nat 
C ) g ) ) )  ->  (
v  e.  ( ( A  Func  C )  X.  ( A  Func  C
) )  /\  h  e.  ( A  Func  C
) ) )
4140simpld 445 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( ph  /\  ( v  e.  ( ( A  Func  C
)  X.  ( A 
Func  C ) )  /\  h  e.  ( A  Func  C ) ) )  /\  f  =  ( 1st `  v ) )  /\  g  =  ( 2nd `  v
) )  /\  (
b  e.  ( g ( A Nat  C ) h )  /\  a  e.  ( f ( A Nat 
C ) g ) ) )  ->  v  e.  ( ( A  Func  C )  X.  ( A 
Func  C ) ) )
42 xp1st 6165 . . . . . . . . . . . . . . . . 17  |-  ( v  e.  ( ( A 
Func  C )  X.  ( A  Func  C ) )  ->  ( 1st `  v
)  e.  ( A 
Func  C ) )
4341, 42syl 15 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( ph  /\  ( v  e.  ( ( A  Func  C
)  X.  ( A 
Func  C ) )  /\  h  e.  ( A  Func  C ) ) )  /\  f  =  ( 1st `  v ) )  /\  g  =  ( 2nd `  v
) )  /\  (
b  e.  ( g ( A Nat  C ) h )  /\  a  e.  ( f ( A Nat 
C ) g ) ) )  ->  ( 1st `  v )  e.  ( A  Func  C
) )
4439, 43eqeltrd 2370 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ph  /\  ( v  e.  ( ( A  Func  C
)  X.  ( A 
Func  C ) )  /\  h  e.  ( A  Func  C ) ) )  /\  f  =  ( 1st `  v ) )  /\  g  =  ( 2nd `  v
) )  /\  (
b  e.  ( g ( A Nat  C ) h )  /\  a  e.  ( f ( A Nat 
C ) g ) ) )  ->  f  e.  ( A  Func  C
) )
45 1st2ndbr 6185 . . . . . . . . . . . . . . 15  |-  ( ( Rel  ( A  Func  C )  /\  f  e.  ( A  Func  C
) )  ->  ( 1st `  f ) ( A  Func  C )
( 2nd `  f
) )
4638, 44, 45sylancr 644 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ph  /\  ( v  e.  ( ( A  Func  C
)  X.  ( A 
Func  C ) )  /\  h  e.  ( A  Func  C ) ) )  /\  f  =  ( 1st `  v ) )  /\  g  =  ( 2nd `  v
) )  /\  (
b  e.  ( g ( A Nat  C ) h )  /\  a  e.  ( f ( A Nat 
C ) g ) ) )  ->  ( 1st `  f ) ( A  Func  C )
( 2nd `  f
) )
4737, 31, 46funcf1 13756 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ph  /\  ( v  e.  ( ( A  Func  C
)  X.  ( A 
Func  C ) )  /\  h  e.  ( A  Func  C ) ) )  /\  f  =  ( 1st `  v ) )  /\  g  =  ( 2nd `  v
) )  /\  (
b  e.  ( g ( A Nat  C ) h )  /\  a  e.  ( f ( A Nat 
C ) g ) ) )  ->  ( 1st `  f ) : ( Base `  A
) --> ( Base `  C
) )
4847ffvelrnda 5681 . . . . . . . . . . . 12  |-  ( ( ( ( ( (
ph  /\  ( v  e.  ( ( A  Func  C )  X.  ( A 
Func  C ) )  /\  h  e.  ( A  Func  C ) ) )  /\  f  =  ( 1st `  v ) )  /\  g  =  ( 2nd `  v
) )  /\  (
b  e.  ( g ( A Nat  C ) h )  /\  a  e.  ( f ( A Nat 
C ) g ) ) )  /\  x  e.  ( Base `  A
) )  ->  (
( 1st `  f
) `  x )  e.  ( Base `  C
) )
49 simplr 731 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( ph  /\  ( v  e.  ( ( A  Func  C
)  X.  ( A 
Func  C ) )  /\  h  e.  ( A  Func  C ) ) )  /\  f  =  ( 1st `  v ) )  /\  g  =  ( 2nd `  v
) )  /\  (
b  e.  ( g ( A Nat  C ) h )  /\  a  e.  ( f ( A Nat 
C ) g ) ) )  ->  g  =  ( 2nd `  v
) )
50 xp2nd 6166 . . . . . . . . . . . . . . . . 17  |-  ( v  e.  ( ( A 
Func  C )  X.  ( A  Func  C ) )  ->  ( 2nd `  v
)  e.  ( A 
Func  C ) )
5141, 50syl 15 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( ph  /\  ( v  e.  ( ( A  Func  C
)  X.  ( A 
Func  C ) )  /\  h  e.  ( A  Func  C ) ) )  /\  f  =  ( 1st `  v ) )  /\  g  =  ( 2nd `  v
) )  /\  (
b  e.  ( g ( A Nat  C ) h )  /\  a  e.  ( f ( A Nat 
C ) g ) ) )  ->  ( 2nd `  v )  e.  ( A  Func  C
) )
5249, 51eqeltrd 2370 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ph  /\  ( v  e.  ( ( A  Func  C
)  X.  ( A 
Func  C ) )  /\  h  e.  ( A  Func  C ) ) )  /\  f  =  ( 1st `  v ) )  /\  g  =  ( 2nd `  v
) )  /\  (
b  e.  ( g ( A Nat  C ) h )  /\  a  e.  ( f ( A Nat 
C ) g ) ) )  ->  g  e.  ( A  Func  C
) )
53 1st2ndbr 6185 . . . . . . . . . . . . . . 15  |-  ( ( Rel  ( A  Func  C )  /\  g  e.  ( A  Func  C
) )  ->  ( 1st `  g ) ( A  Func  C )
( 2nd `  g
) )
5438, 52, 53sylancr 644 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ph  /\  ( v  e.  ( ( A  Func  C
)  X.  ( A 
Func  C ) )  /\  h  e.  ( A  Func  C ) ) )  /\  f  =  ( 1st `  v ) )  /\  g  =  ( 2nd `  v
) )  /\  (
b  e.  ( g ( A Nat  C ) h )  /\  a  e.  ( f ( A Nat 
C ) g ) ) )  ->  ( 1st `  g ) ( A  Func  C )
( 2nd `  g
) )
5537, 31, 54funcf1 13756 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ph  /\  ( v  e.  ( ( A  Func  C
)  X.  ( A 
Func  C ) )  /\  h  e.  ( A  Func  C ) ) )  /\  f  =  ( 1st `  v ) )  /\  g  =  ( 2nd `  v
) )  /\  (
b  e.  ( g ( A Nat  C ) h )  /\  a  e.  ( f ( A Nat 
C ) g ) ) )  ->  ( 1st `  g ) : ( Base `  A
) --> ( Base `  C
) )
5655ffvelrnda 5681 . . . . . . . . . . . 12  |-  ( ( ( ( ( (
ph  /\  ( v  e.  ( ( A  Func  C )  X.  ( A 
Func  C ) )  /\  h  e.  ( A  Func  C ) ) )  /\  f  =  ( 1st `  v ) )  /\  g  =  ( 2nd `  v
) )  /\  (
b  e.  ( g ( A Nat  C ) h )  /\  a  e.  ( f ( A Nat 
C ) g ) ) )  /\  x  e.  ( Base `  A
) )  ->  (
( 1st `  g
) `  x )  e.  ( Base `  C
) )
5740simprd 449 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ph  /\  ( v  e.  ( ( A  Func  C
)  X.  ( A 
Func  C ) )  /\  h  e.  ( A  Func  C ) ) )  /\  f  =  ( 1st `  v ) )  /\  g  =  ( 2nd `  v
) )  /\  (
b  e.  ( g ( A Nat  C ) h )  /\  a  e.  ( f ( A Nat 
C ) g ) ) )  ->  h  e.  ( A  Func  C
) )
58 1st2ndbr 6185 . . . . . . . . . . . . . . 15  |-  ( ( Rel  ( A  Func  C )  /\  h  e.  ( A  Func  C
) )  ->  ( 1st `  h ) ( A  Func  C )
( 2nd `  h
) )
5938, 57, 58sylancr 644 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ph  /\  ( v  e.  ( ( A  Func  C
)  X.  ( A 
Func  C ) )  /\  h  e.  ( A  Func  C ) ) )  /\  f  =  ( 1st `  v ) )  /\  g  =  ( 2nd `  v
) )  /\  (
b  e.  ( g ( A Nat  C ) h )  /\  a  e.  ( f ( A Nat 
C ) g ) ) )  ->  ( 1st `  h ) ( A  Func  C )
( 2nd `  h
) )
6037, 31, 59funcf1 13756 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ph  /\  ( v  e.  ( ( A  Func  C
)  X.  ( A 
Func  C ) )  /\  h  e.  ( A  Func  C ) ) )  /\  f  =  ( 1st `  v ) )  /\  g  =  ( 2nd `  v
) )  /\  (
b  e.  ( g ( A Nat  C ) h )  /\  a  e.  ( f ( A Nat 
C ) g ) ) )  ->  ( 1st `  h ) : ( Base `  A
) --> ( Base `  C
) )
6160ffvelrnda 5681 . . . . . . . . . . . 12  |-  ( ( ( ( ( (
ph  /\  ( v  e.  ( ( A  Func  C )  X.  ( A 
Func  C ) )  /\  h  e.  ( A  Func  C ) ) )  /\  f  =  ( 1st `  v ) )  /\  g  =  ( 2nd `  v
) )  /\  (
b  e.  ( g ( A Nat  C ) h )  /\  a  e.  ( f ( A Nat 
C ) g ) ) )  /\  x  e.  ( Base `  A
) )  ->  (
( 1st `  h
) `  x )  e.  ( Base `  C
) )
62 eqid 2296 . . . . . . . . . . . . 13  |-  ( A Nat 
C )  =  ( A Nat  C )
63 simplrr 737 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( (
ph  /\  ( v  e.  ( ( A  Func  C )  X.  ( A 
Func  C ) )  /\  h  e.  ( A  Func  C ) ) )  /\  f  =  ( 1st `  v ) )  /\  g  =  ( 2nd `  v
) )  /\  (
b  e.  ( g ( A Nat  C ) h )  /\  a  e.  ( f ( A Nat 
C ) g ) ) )  /\  x  e.  ( Base `  A
) )  ->  a  e.  ( f ( A Nat 
C ) g ) )
6462, 63nat1st2nd 13841 . . . . . . . . . . . . 13  |-  ( ( ( ( ( (
ph  /\  ( v  e.  ( ( A  Func  C )  X.  ( A 
Func  C ) )  /\  h  e.  ( A  Func  C ) ) )  /\  f  =  ( 1st `  v ) )  /\  g  =  ( 2nd `  v
) )  /\  (
b  e.  ( g ( A Nat  C ) h )  /\  a  e.  ( f ( A Nat 
C ) g ) ) )  /\  x  e.  ( Base `  A
) )  ->  a  e.  ( <. ( 1st `  f
) ,  ( 2nd `  f ) >. ( A Nat  C ) <. ( 1st `  g ) ,  ( 2nd `  g
) >. ) )
65 simpr 447 . . . . . . . . . . . . 13  |-  ( ( ( ( ( (
ph  /\  ( v  e.  ( ( A  Func  C )  X.  ( A 
Func  C ) )  /\  h  e.  ( A  Func  C ) ) )  /\  f  =  ( 1st `  v ) )  /\  g  =  ( 2nd `  v
) )  /\  (
b  e.  ( g ( A Nat  C ) h )  /\  a  e.  ( f ( A Nat 
C ) g ) ) )  /\  x  e.  ( Base `  A
) )  ->  x  e.  ( Base `  A
) )
6662, 64, 37, 32, 65natcl 13843 . . . . . . . . . . . 12  |-  ( ( ( ( ( (
ph  /\  ( v  e.  ( ( A  Func  C )  X.  ( A 
Func  C ) )  /\  h  e.  ( A  Func  C ) ) )  /\  f  =  ( 1st `  v ) )  /\  g  =  ( 2nd `  v
) )  /\  (
b  e.  ( g ( A Nat  C ) h )  /\  a  e.  ( f ( A Nat 
C ) g ) ) )  /\  x  e.  ( Base `  A
) )  ->  (
a `  x )  e.  ( ( ( 1st `  f ) `  x
) (  Hom  `  C
) ( ( 1st `  g ) `  x
) ) )
67 simplrl 736 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( (
ph  /\  ( v  e.  ( ( A  Func  C )  X.  ( A 
Func  C ) )  /\  h  e.  ( A  Func  C ) ) )  /\  f  =  ( 1st `  v ) )  /\  g  =  ( 2nd `  v
) )  /\  (
b  e.  ( g ( A Nat  C ) h )  /\  a  e.  ( f ( A Nat 
C ) g ) ) )  /\  x  e.  ( Base `  A
) )  ->  b  e.  ( g ( A Nat 
C ) h ) )
6862, 67nat1st2nd 13841 . . . . . . . . . . . . 13  |-  ( ( ( ( ( (
ph  /\  ( v  e.  ( ( A  Func  C )  X.  ( A 
Func  C ) )  /\  h  e.  ( A  Func  C ) ) )  /\  f  =  ( 1st `  v ) )  /\  g  =  ( 2nd `  v
) )  /\  (
b  e.  ( g ( A Nat  C ) h )  /\  a  e.  ( f ( A Nat 
C ) g ) ) )  /\  x  e.  ( Base `  A
) )  ->  b  e.  ( <. ( 1st `  g
) ,  ( 2nd `  g ) >. ( A Nat  C ) <. ( 1st `  h ) ,  ( 2nd `  h
) >. ) )
6962, 68, 37, 32, 65natcl 13843 . . . . . . . . . . . 12  |-  ( ( ( ( ( (
ph  /\  ( v  e.  ( ( A  Func  C )  X.  ( A 
Func  C ) )  /\  h  e.  ( A  Func  C ) ) )  /\  f  =  ( 1st `  v ) )  /\  g  =  ( 2nd `  v
) )  /\  (
b  e.  ( g ( A Nat  C ) h )  /\  a  e.  ( f ( A Nat 
C ) g ) ) )  /\  x  e.  ( Base `  A
) )  ->  (
b `  x )  e.  ( ( ( 1st `  g ) `  x
) (  Hom  `  C
) ( ( 1st `  h ) `  x
) ) )
7031, 32, 33, 34, 35, 36, 48, 56, 61, 66, 69comfeqval 13627 . . . . . . . . . . 11  |-  ( ( ( ( ( (
ph  /\  ( v  e.  ( ( A  Func  C )  X.  ( A 
Func  C ) )  /\  h  e.  ( A  Func  C ) ) )  /\  f  =  ( 1st `  v ) )  /\  g  =  ( 2nd `  v
) )  /\  (
b  e.  ( g ( A Nat  C ) h )  /\  a  e.  ( f ( A Nat 
C ) g ) ) )  /\  x  e.  ( Base `  A
) )  ->  (
( b `  x
) ( <. (
( 1st `  f
) `  x ) ,  ( ( 1st `  g ) `  x
) >. (comp `  C
) ( ( 1st `  h ) `  x
) ) ( a `
 x ) )  =  ( ( b `
 x ) (
<. ( ( 1st `  f
) `  x ) ,  ( ( 1st `  g ) `  x
) >. (comp `  D
) ( ( 1st `  h ) `  x
) ) ( a `
 x ) ) )
7130, 70mpteq12dva 4113 . . . . . . . . . 10  |-  ( ( ( ( ( ph  /\  ( v  e.  ( ( A  Func  C
)  X.  ( A 
Func  C ) )  /\  h  e.  ( A  Func  C ) ) )  /\  f  =  ( 1st `  v ) )  /\  g  =  ( 2nd `  v
) )  /\  (
b  e.  ( g ( A Nat  C ) h )  /\  a  e.  ( f ( A Nat 
C ) g ) ) )  ->  (
x  e.  ( Base `  A )  |->  ( ( b `  x ) ( <. ( ( 1st `  f ) `  x
) ,  ( ( 1st `  g ) `
 x ) >.
(comp `  C )
( ( 1st `  h
) `  x )
) ( a `  x ) ) )  =  ( x  e.  ( Base `  B
)  |->  ( ( b `
 x ) (
<. ( ( 1st `  f
) `  x ) ,  ( ( 1st `  g ) `  x
) >. (comp `  D
) ( ( 1st `  h ) `  x
) ) ( a `
 x ) ) ) )
7226, 28, 71mpt2eq123dva 5925 . . . . . . . . 9  |-  ( ( ( ( ph  /\  ( v  e.  ( ( A  Func  C
)  X.  ( A 
Func  C ) )  /\  h  e.  ( A  Func  C ) ) )  /\  f  =  ( 1st `  v ) )  /\  g  =  ( 2nd `  v
) )  ->  (
b  e.  ( g ( A Nat  C ) h ) ,  a  e.  ( f ( A Nat  C ) g )  |->  ( x  e.  ( Base `  A
)  |->  ( ( b `
 x ) (
<. ( ( 1st `  f
) `  x ) ,  ( ( 1st `  g ) `  x
) >. (comp `  C
) ( ( 1st `  h ) `  x
) ) ( a `
 x ) ) ) )  =  ( b  e.  ( g ( B Nat  D ) h ) ,  a  e.  ( f ( B Nat  D ) g )  |->  ( x  e.  ( Base `  B
)  |->  ( ( b `
 x ) (
<. ( ( 1st `  f
) `  x ) ,  ( ( 1st `  g ) `  x
) >. (comp `  D
) ( ( 1st `  h ) `  x
) ) ( a `
 x ) ) ) ) )
73 csbeq1a 3102 . . . . . . . . . 10  |-  ( g  =  ( 2nd `  v
)  ->  ( b  e.  ( g ( B Nat 
D ) h ) ,  a  e.  ( f ( B Nat  D
) g )  |->  ( x  e.  ( Base `  B )  |->  ( ( b `  x ) ( <. ( ( 1st `  f ) `  x
) ,  ( ( 1st `  g ) `
 x ) >.
(comp `  D )
( ( 1st `  h
) `  x )
) ( a `  x ) ) ) )  =  [_ ( 2nd `  v )  / 
g ]_ ( b  e.  ( g ( B Nat 
D ) h ) ,  a  e.  ( f ( B Nat  D
) g )  |->  ( x  e.  ( Base `  B )  |->  ( ( b `  x ) ( <. ( ( 1st `  f ) `  x
) ,  ( ( 1st `  g ) `
 x ) >.
(comp `  D )
( ( 1st `  h
) `  x )
) ( a `  x ) ) ) ) )
7473adantl 452 . . . . . . . . 9  |-  ( ( ( ( ph  /\  ( v  e.  ( ( A  Func  C
)  X.  ( A 
Func  C ) )  /\  h  e.  ( A  Func  C ) ) )  /\  f  =  ( 1st `  v ) )  /\  g  =  ( 2nd `  v
) )  ->  (
b  e.  ( g ( B Nat  D ) h ) ,  a  e.  ( f ( B Nat  D ) g )  |->  ( x  e.  ( Base `  B
)  |->  ( ( b `
 x ) (
<. ( ( 1st `  f
) `  x ) ,  ( ( 1st `  g ) `  x
) >. (comp `  D
) ( ( 1st `  h ) `  x
) ) ( a `
 x ) ) ) )  =  [_ ( 2nd `  v )  /  g ]_ (
b  e.  ( g ( B Nat  D ) h ) ,  a  e.  ( f ( B Nat  D ) g )  |->  ( x  e.  ( Base `  B
)  |->  ( ( b `
 x ) (
<. ( ( 1st `  f
) `  x ) ,  ( ( 1st `  g ) `  x
) >. (comp `  D
) ( ( 1st `  h ) `  x
) ) ( a `
 x ) ) ) ) )
7572, 74eqtrd 2328 . . . . . . . 8  |-  ( ( ( ( ph  /\  ( v  e.  ( ( A  Func  C
)  X.  ( A 
Func  C ) )  /\  h  e.  ( A  Func  C ) ) )  /\  f  =  ( 1st `  v ) )  /\  g  =  ( 2nd `  v
) )  ->  (
b  e.  ( g ( A Nat  C ) h ) ,  a  e.  ( f ( A Nat  C ) g )  |->  ( x  e.  ( Base `  A
)  |->  ( ( b `
 x ) (
<. ( ( 1st `  f
) `  x ) ,  ( ( 1st `  g ) `  x
) >. (comp `  C
) ( ( 1st `  h ) `  x
) ) ( a `
 x ) ) ) )  =  [_ ( 2nd `  v )  /  g ]_ (
b  e.  ( g ( B Nat  D ) h ) ,  a  e.  ( f ( B Nat  D ) g )  |->  ( x  e.  ( Base `  B
)  |->  ( ( b `
 x ) (
<. ( ( 1st `  f
) `  x ) ,  ( ( 1st `  g ) `  x
) >. (comp `  D
) ( ( 1st `  h ) `  x
) ) ( a `
 x ) ) ) ) )
7620, 22, 24, 75csbiedf 3131 . . . . . . 7  |-  ( ( ( ph  /\  (
v  e.  ( ( A  Func  C )  X.  ( A  Func  C
) )  /\  h  e.  ( A  Func  C
) ) )  /\  f  =  ( 1st `  v ) )  ->  [_ ( 2nd `  v
)  /  g ]_ ( b  e.  ( g ( A Nat  C
) h ) ,  a  e.  ( f ( A Nat  C ) g )  |->  ( x  e.  ( Base `  A
)  |->  ( ( b `
 x ) (
<. ( ( 1st `  f
) `  x ) ,  ( ( 1st `  g ) `  x
) >. (comp `  C
) ( ( 1st `  h ) `  x
) ) ( a `
 x ) ) ) )  =  [_ ( 2nd `  v )  /  g ]_ (
b  e.  ( g ( B Nat  D ) h ) ,  a  e.  ( f ( B Nat  D ) g )  |->  ( x  e.  ( Base `  B
)  |->  ( ( b `
 x ) (
<. ( ( 1st `  f
) `  x ) ,  ( ( 1st `  g ) `  x
) >. (comp `  D
) ( ( 1st `  h ) `  x
) ) ( a `
 x ) ) ) ) )
77 csbeq1a 3102 . . . . . . . 8  |-  ( f  =  ( 1st `  v
)  ->  [_ ( 2nd `  v )  /  g ]_ ( b  e.  ( g ( B Nat  D
) h ) ,  a  e.  ( f ( B Nat  D ) g )  |->  ( x  e.  ( Base `  B
)  |->  ( ( b `
 x ) (
<. ( ( 1st `  f
) `  x ) ,  ( ( 1st `  g ) `  x
) >. (comp `  D
) ( ( 1st `  h ) `  x
) ) ( a `
 x ) ) ) )  =  [_ ( 1st `  v )  /  f ]_ [_ ( 2nd `  v )  / 
g ]_ ( b  e.  ( g ( B Nat 
D ) h ) ,  a  e.  ( f ( B Nat  D
) g )  |->  ( x  e.  ( Base `  B )  |->  ( ( b `  x ) ( <. ( ( 1st `  f ) `  x
) ,  ( ( 1st `  g ) `
 x ) >.
(comp `  D )
( ( 1st `  h
) `  x )
) ( a `  x ) ) ) ) )
7877adantl 452 . . . . . . 7  |-  ( ( ( ph  /\  (
v  e.  ( ( A  Func  C )  X.  ( A  Func  C
) )  /\  h  e.  ( A  Func  C
) ) )  /\  f  =  ( 1st `  v ) )  ->  [_ ( 2nd `  v
)  /  g ]_ ( b  e.  ( g ( B Nat  D
) h ) ,  a  e.  ( f ( B Nat  D ) g )  |->  ( x  e.  ( Base `  B
)  |->  ( ( b `
 x ) (
<. ( ( 1st `  f
) `  x ) ,  ( ( 1st `  g ) `  x
) >. (comp `  D
) ( ( 1st `  h ) `  x
) ) ( a `
 x ) ) ) )  =  [_ ( 1st `  v )  /  f ]_ [_ ( 2nd `  v )  / 
g ]_ ( b  e.  ( g ( B Nat 
D ) h ) ,  a  e.  ( f ( B Nat  D
) g )  |->  ( x  e.  ( Base `  B )  |->  ( ( b `  x ) ( <. ( ( 1st `  f ) `  x
) ,  ( ( 1st `  g ) `
 x ) >.
(comp `  D )
( ( 1st `  h
) `  x )
) ( a `  x ) ) ) ) )
7976, 78eqtrd 2328 . . . . . 6  |-  ( ( ( ph  /\  (
v  e.  ( ( A  Func  C )  X.  ( A  Func  C
) )  /\  h  e.  ( A  Func  C
) ) )  /\  f  =  ( 1st `  v ) )  ->  [_ ( 2nd `  v
)  /  g ]_ ( b  e.  ( g ( A Nat  C
) h ) ,  a  e.  ( f ( A Nat  C ) g )  |->  ( x  e.  ( Base `  A
)  |->  ( ( b `
 x ) (
<. ( ( 1st `  f
) `  x ) ,  ( ( 1st `  g ) `  x
) >. (comp `  C
) ( ( 1st `  h ) `  x
) ) ( a `
 x ) ) ) )  =  [_ ( 1st `  v )  /  f ]_ [_ ( 2nd `  v )  / 
g ]_ ( b  e.  ( g ( B Nat 
D ) h ) ,  a  e.  ( f ( B Nat  D
) g )  |->  ( x  e.  ( Base `  B )  |->  ( ( b `  x ) ( <. ( ( 1st `  f ) `  x
) ,  ( ( 1st `  g ) `
 x ) >.
(comp `  D )
( ( 1st `  h
) `  x )
) ( a `  x ) ) ) ) )
8015, 17, 19, 79csbiedf 3131 . . . . 5  |-  ( (
ph  /\  ( v  e.  ( ( A  Func  C )  X.  ( A 
Func  C ) )  /\  h  e.  ( A  Func  C ) ) )  ->  [_ ( 1st `  v
)  /  f ]_ [_ ( 2nd `  v
)  /  g ]_ ( b  e.  ( g ( A Nat  C
) h ) ,  a  e.  ( f ( A Nat  C ) g )  |->  ( x  e.  ( Base `  A
)  |->  ( ( b `
 x ) (
<. ( ( 1st `  f
) `  x ) ,  ( ( 1st `  g ) `  x
) >. (comp `  C
) ( ( 1st `  h ) `  x
) ) ( a `
 x ) ) ) )  =  [_ ( 1st `  v )  /  f ]_ [_ ( 2nd `  v )  / 
g ]_ ( b  e.  ( g ( B Nat 
D ) h ) ,  a  e.  ( f ( B Nat  D
) g )  |->  ( x  e.  ( Base `  B )  |->  ( ( b `  x ) ( <. ( ( 1st `  f ) `  x
) ,  ( ( 1st `  g ) `
 x ) >.
(comp `  D )
( ( 1st `  h
) `  x )
) ( a `  x ) ) ) ) )
8113, 14, 80mpt2eq123dva 5925 . . . 4  |-  ( ph  ->  ( v  e.  ( ( A  Func  C
)  X.  ( A 
Func  C ) ) ,  h  e.  ( A 
Func  C )  |->  [_ ( 1st `  v )  / 
f ]_ [_ ( 2nd `  v )  /  g ]_ ( b  e.  ( g ( A Nat  C
) h ) ,  a  e.  ( f ( A Nat  C ) g )  |->  ( x  e.  ( Base `  A
)  |->  ( ( b `
 x ) (
<. ( ( 1st `  f
) `  x ) ,  ( ( 1st `  g ) `  x
) >. (comp `  C
) ( ( 1st `  h ) `  x
) ) ( a `
 x ) ) ) ) )  =  ( v  e.  ( ( B  Func  D
)  X.  ( B 
Func  D ) ) ,  h  e.  ( B 
Func  D )  |->  [_ ( 1st `  v )  / 
f ]_ [_ ( 2nd `  v )  /  g ]_ ( b  e.  ( g ( B Nat  D
) h ) ,  a  e.  ( f ( B Nat  D ) g )  |->  ( x  e.  ( Base `  B
)  |->  ( ( b `
 x ) (
<. ( ( 1st `  f
) `  x ) ,  ( ( 1st `  g ) `  x
) >. (comp `  D
) ( ( 1st `  h ) `  x
) ) ( a `
 x ) ) ) ) ) )
8281opeq2d 3819 . . 3  |-  ( ph  -> 
<. (comp `  ndx ) ,  ( v  e.  ( ( A  Func  C
)  X.  ( A 
Func  C ) ) ,  h  e.  ( A 
Func  C )  |->  [_ ( 1st `  v )  / 
f ]_ [_ ( 2nd `  v )  /  g ]_ ( b  e.  ( g ( A Nat  C
) h ) ,  a  e.  ( f ( A Nat  C ) g )  |->  ( x  e.  ( Base `  A
)  |->  ( ( b `
 x ) (
<. ( ( 1st `  f
) `  x ) ,  ( ( 1st `  g ) `  x
) >. (comp `  C
) ( ( 1st `  h ) `  x
) ) ( a `
 x ) ) ) ) ) >.  =  <. (comp `  ndx ) ,  ( v  e.  ( ( B  Func  D )  X.  ( B 
Func  D ) ) ,  h  e.  ( B 
Func  D )  |->  [_ ( 1st `  v )  / 
f ]_ [_ ( 2nd `  v )  /  g ]_ ( b  e.  ( g ( B Nat  D
) h ) ,  a  e.  ( f ( B Nat  D ) g )  |->  ( x  e.  ( Base `  B
)  |->  ( ( b `
 x ) (
<. ( ( 1st `  f
) `  x ) ,  ( ( 1st `  g ) `  x
) >. (comp `  D
) ( ( 1st `  h ) `  x
) ) ( a `
 x ) ) ) ) ) >.
)
8310, 12, 82tpeq123d 3734 . 2  |-  ( ph  ->  { <. ( Base `  ndx ) ,  ( A  Func  C ) >. ,  <. (  Hom  `  ndx ) ,  ( A Nat  C )
>. ,  <. (comp `  ndx ) ,  ( v  e.  ( ( A 
Func  C )  X.  ( A  Func  C ) ) ,  h  e.  ( A  Func  C )  |-> 
[_ ( 1st `  v
)  /  f ]_ [_ ( 2nd `  v
)  /  g ]_ ( b  e.  ( g ( A Nat  C
) h ) ,  a  e.  ( f ( A Nat  C ) g )  |->  ( x  e.  ( Base `  A
)  |->  ( ( b `
 x ) (
<. ( ( 1st `  f
) `  x ) ,  ( ( 1st `  g ) `  x
) >. (comp `  C
) ( ( 1st `  h ) `  x
) ) ( a `
 x ) ) ) ) ) >. }  =  { <. ( Base `  ndx ) ,  ( B  Func  D
) >. ,  <. (  Hom  `  ndx ) ,  ( B Nat  D )
>. ,  <. (comp `  ndx ) ,  ( v  e.  ( ( B 
Func  D )  X.  ( B  Func  D ) ) ,  h  e.  ( B  Func  D )  |-> 
[_ ( 1st `  v
)  /  f ]_ [_ ( 2nd `  v
)  /  g ]_ ( b  e.  ( g ( B Nat  D
) h ) ,  a  e.  ( f ( B Nat  D ) g )  |->  ( x  e.  ( Base `  B
)  |->  ( ( b `
 x ) (
<. ( ( 1st `  f
) `  x ) ,  ( ( 1st `  g ) `  x
) >. (comp `  D
) ( ( 1st `  h ) `  x
) ) ( a `
 x ) ) ) ) ) >. } )
84 eqid 2296 . . 3  |-  ( A FuncCat  C )  =  ( A FuncCat  C )
85 eqid 2296 . . 3  |-  ( A 
Func  C )  =  ( A  Func  C )
86 eqidd 2297 . . 3  |-  ( ph  ->  ( v  e.  ( ( A  Func  C
)  X.  ( A 
Func  C ) ) ,  h  e.  ( A 
Func  C )  |->  [_ ( 1st `  v )  / 
f ]_ [_ ( 2nd `  v )  /  g ]_ ( b  e.  ( g ( A Nat  C
) h ) ,  a  e.  ( f ( A Nat  C ) g )  |->  ( x  e.  ( Base `  A
)  |->  ( ( b `
 x ) (
<. ( ( 1st `  f
) `  x ) ,  ( ( 1st `  g ) `  x
) >. (comp `  C
) ( ( 1st `  h ) `  x
) ) ( a `
 x ) ) ) ) )  =  ( v  e.  ( ( A  Func  C
)  X.  ( A 
Func  C ) ) ,  h  e.  ( A 
Func  C )  |->  [_ ( 1st `  v )  / 
f ]_ [_ ( 2nd `  v )  /  g ]_ ( b  e.  ( g ( A Nat  C
) h ) ,  a  e.  ( f ( A Nat  C ) g )  |->  ( x  e.  ( Base `  A
)  |->  ( ( b `
 x ) (
<. ( ( 1st `  f
) `  x ) ,  ( ( 1st `  g ) `  x
) >. (comp `  C
) ( ( 1st `  h ) `  x
) ) ( a `
 x ) ) ) ) ) )
8784, 85, 62, 37, 33, 5, 7, 86fucval 13848 . 2  |-  ( ph  ->  ( A FuncCat  C )  =  { <. ( Base `  ndx ) ,  ( A  Func  C ) >. ,  <. (  Hom  `  ndx ) ,  ( A Nat  C )
>. ,  <. (comp `  ndx ) ,  ( v  e.  ( ( A 
Func  C )  X.  ( A  Func  C ) ) ,  h  e.  ( A  Func  C )  |-> 
[_ ( 1st `  v
)  /  f ]_ [_ ( 2nd `  v
)  /  g ]_ ( b  e.  ( g ( A Nat  C
) h ) ,  a  e.  ( f ( A Nat  C ) g )  |->  ( x  e.  ( Base `  A
)  |->  ( ( b `
 x ) (
<. ( ( 1st `  f
) `  x ) ,  ( ( 1st `  g ) `  x
) >. (comp `  C
) ( ( 1st `  h ) `  x
) ) ( a `
 x ) ) ) ) ) >. } )
88 eqid 2296 . . 3  |-  ( B FuncCat  D )  =  ( B FuncCat  D )
89 eqid 2296 . . 3  |-  ( B 
Func  D )  =  ( B  Func  D )
90 eqid 2296 . . 3  |-  ( B Nat 
D )  =  ( B Nat  D )
91 eqid 2296 . . 3  |-  ( Base `  B )  =  (
Base `  B )
92 eqidd 2297 . . 3  |-  ( ph  ->  ( v  e.  ( ( B  Func  D
)  X.  ( B 
Func  D ) ) ,  h  e.  ( B 
Func  D )  |->  [_ ( 1st `  v )  / 
f ]_ [_ ( 2nd `  v )  /  g ]_ ( b  e.  ( g ( B Nat  D
) h ) ,  a  e.  ( f ( B Nat  D ) g )  |->  ( x  e.  ( Base `  B
)  |->  ( ( b `
 x ) (
<. ( ( 1st `  f
) `  x ) ,  ( ( 1st `  g ) `  x
) >. (comp `  D
) ( ( 1st `  h ) `  x
) ) ( a `
 x ) ) ) ) )  =  ( v  e.  ( ( B  Func  D
)  X.  ( B 
Func  D ) ) ,  h  e.  ( B 
Func  D )  |->  [_ ( 1st `  v )  / 
f ]_ [_ ( 2nd `  v )  /  g ]_ ( b  e.  ( g ( B Nat  D
) h ) ,  a  e.  ( f ( B Nat  D ) g )  |->  ( x  e.  ( Base `  B
)  |->  ( ( b `
 x ) (
<. ( ( 1st `  f
) `  x ) ,  ( ( 1st `  g ) `  x
) >. (comp `  D
) ( ( 1st `  h ) `  x
) ) ( a `
 x ) ) ) ) ) )
9388, 89, 90, 91, 34, 6, 8, 92fucval 13848 . 2  |-  ( ph  ->  ( B FuncCat  D )  =  { <. ( Base `  ndx ) ,  ( B  Func  D ) >. ,  <. (  Hom  `  ndx ) ,  ( B Nat  D )
>. ,  <. (comp `  ndx ) ,  ( v  e.  ( ( B 
Func  D )  X.  ( B  Func  D ) ) ,  h  e.  ( B  Func  D )  |-> 
[_ ( 1st `  v
)  /  f ]_ [_ ( 2nd `  v
)  /  g ]_ ( b  e.  ( g ( B Nat  D
) h ) ,  a  e.  ( f ( B Nat  D ) g )  |->  ( x  e.  ( Base `  B
)  |->  ( ( b `
 x ) (
<. ( ( 1st `  f
) `  x ) ,  ( ( 1st `  g ) `  x
) >. (comp `  D
) ( ( 1st `  h ) `  x
) ) ( a `
 x ) ) ) ) ) >. } )
9483, 87, 933eqtr4d 2338 1  |-  ( ph  ->  ( A FuncCat  C )  =  ( B FuncCat  D
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   F/_wnfc 2419   _Vcvv 2801   [_csb 3094   {ctp 3655   <.cop 3656   class class class wbr 4039    e. cmpt 4093    X. cxp 4703   Rel wrel 4710   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876   1stc1st 6136   2ndc2nd 6137   ndxcnx 13161   Basecbs 13164    Hom chom 13235  compcco 13236   Catccat 13582    Homf chomf 13584  compfccomf 13585    Func cfunc 13744   Nat cnat 13831   FuncCat cfuc 13832
This theorem is referenced by:  oyoncl  14060
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-map 6790  df-ixp 6834  df-cat 13586  df-cid 13587  df-homf 13588  df-comf 13589  df-func 13748  df-nat 13833  df-fuc 13834
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