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

Theorem fucpropd 13851
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 13774 . . . 4  |-  ( ph  ->  ( A  Func  C
)  =  ( B 
Func  D ) )
109opeq2d 3803 . . 3  |-  ( ph  -> 
<. ( Base `  ndx ) ,  ( A  Func  C ) >.  =  <. (
Base `  ndx ) ,  ( B  Func  D
) >. )
111, 2, 3, 4, 5, 6, 7, 8natpropd 13850 . . . 4  |-  ( ph  ->  ( A Nat  C )  =  ( B Nat  D
) )
1211opeq2d 3803 . . 3  |-  ( ph  -> 
<. (  Hom  `  ndx ) ,  ( A Nat  C ) >.  =  <. (  Hom  `  ndx ) ,  ( B Nat  D )
>. )
139, 9xpeq12d 4714 . . . . 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 1605 . . . . . 6  |-  F/ f ( ph  /\  (
v  e.  ( ( A  Func  C )  X.  ( A  Func  C
) )  /\  h  e.  ( A  Func  C
) ) )
16 nfcsb1v 3113 . . . . . . 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 5539 . . . . . . 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 1605 . . . . . . . 8  |-  F/ g ( ( ph  /\  ( v  e.  ( ( A  Func  C
)  X.  ( A 
Func  C ) )  /\  h  e.  ( A  Func  C ) ) )  /\  f  =  ( 1st `  v ) )
21 nfcsb1v 3113 . . . . . . . . 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 5539 . . . . . . . . 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 5875 . . . . . . . . . 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 5875 . . . . . . . . . 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 13599 . . . . . . . . . . . 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 2283 . . . . . . . . . . . 12  |-  ( Base `  C )  =  (
Base `  C )
32 eqid 2283 . . . . . . . . . . . 12  |-  (  Hom  `  C )  =  (  Hom  `  C )
33 eqid 2283 . . . . . . . . . . . 12  |-  (comp `  C )  =  (comp `  C )
34 eqid 2283 . . . . . . . . . . . 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 2283 . . . . . . . . . . . . . 14  |-  ( Base `  A )  =  (
Base `  A )
38 relfunc 13736 . . . . . . . . . . . . . . 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 6149 . . . . . . . . . . . . . . . . 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 2357 . . . . . . . . . . . . . . 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 6169 . . . . . . . . . . . . . . 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 13740 . . . . . . . . . . . . 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 5665 . . . . . . . . . . . 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 6150 . . . . . . . . . . . . . . . . 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 2357 . . . . . . . . . . . . . . 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 6169 . . . . . . . . . . . . . . 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 13740 . . . . . . . . . . . . 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 5665 . . . . . . . . . . . 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 6169 . . . . . . . . . . . . . . 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 13740 . . . . . . . . . . . . 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 5665 . . . . . . . . . . . 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 2283 . . . . . . . . . . . . 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 13825 . . . . . . . . . . . . 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 13827 . . . . . . . . . . . 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 13825 . . . . . . . . . . . . 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 13827 . . . . . . . . . . . 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 13611 . . . . . . . . . . 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 4097 . . . . . . . . . 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 5909 . . . . . . . . 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 3089 . . . . . . . . . 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 2315 . . . . . . . 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 3118 . . . . . . 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 3089 . . . . . . . 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 2315 . . . . . 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 3118 . . . . 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 5909 . . . 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 3803 . . 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 3721 . 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 2283 . . 3  |-  ( A FuncCat  C )  =  ( A FuncCat  C )
85 eqid 2283 . . 3  |-  ( A 
Func  C )  =  ( A  Func  C )
86 eqidd 2284 . . 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 13832 . 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 2283 . . 3  |-  ( B FuncCat  D )  =  ( B FuncCat  D )
89 eqid 2283 . . 3  |-  ( B 
Func  D )  =  ( B  Func  D )
90 eqid 2283 . . 3  |-  ( B Nat 
D )  =  ( B Nat  D )
91 eqid 2283 . . 3  |-  ( Base `  B )  =  (
Base `  B )
92 eqidd 2284 . . 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 13832 . 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 2325 1  |-  ( ph  ->  ( A FuncCat  C )  =  ( B FuncCat  D
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   F/_wnfc 2406   _Vcvv 2788   [_csb 3081   {ctp 3642   <.cop 3643   class class class wbr 4023    e. cmpt 4077    X. cxp 4687   Rel wrel 4694   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   1stc1st 6120   2ndc2nd 6121   ndxcnx 13145   Basecbs 13148    Hom chom 13219  compcco 13220   Catccat 13566    Homf chomf 13568  compfccomf 13569    Func cfunc 13728   Nat cnat 13815   FuncCat cfuc 13816
This theorem is referenced by:  oyoncl  14044
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-map 6774  df-ixp 6818  df-cat 13570  df-cid 13571  df-homf 13572  df-comf 13573  df-func 13732  df-nat 13817  df-fuc 13818
  Copyright terms: Public domain W3C validator