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Theorem fucpropd 15393
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  ->  ( Hom f  `  A )  =  ( Hom f  `  B ) )
fucpropd.2  |-  ( ph  ->  (compf `  A )  =  (compf `  B ) )
fucpropd.3  |-  ( ph  ->  ( Hom f  `  C )  =  ( Hom f  `  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  ->  ( Hom f  `  A )  =  ( Hom f  `  B ) )
2 fucpropd.2 . . . . 5  |-  ( ph  ->  (compf `  A )  =  (compf `  B ) )
3 fucpropd.3 . . . . 5  |-  ( ph  ->  ( Hom f  `  C )  =  ( Hom f  `  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 15316 . . . 4  |-  ( ph  ->  ( A  Func  C
)  =  ( B 
Func  D ) )
109opeq2d 4226 . . 3  |-  ( ph  -> 
<. ( Base `  ndx ) ,  ( A  Func  C ) >.  =  <. (
Base `  ndx ) ,  ( B  Func  D
) >. )
111, 2, 3, 4, 5, 6, 7, 8natpropd 15392 . . . 4  |-  ( ph  ->  ( A Nat  C )  =  ( B Nat  D
) )
1211opeq2d 4226 . . 3  |-  ( ph  -> 
<. ( Hom  `  ndx ) ,  ( A Nat  C ) >.  =  <. ( Hom  `  ndx ) ,  ( B Nat  D )
>. )
139sqxpeqd 5034 . . . . 5  |-  ( ph  ->  ( ( A  Func  C )  X.  ( A 
Func  C ) )  =  ( ( B  Func  D )  X.  ( B 
Func  D ) ) )
149adantr 465 . . . . 5  |-  ( (
ph  /\  v  e.  ( ( A  Func  C )  X.  ( A 
Func  C ) ) )  ->  ( A  Func  C )  =  ( B 
Func  D ) )
15 nfv 1708 . . . . . 6  |-  F/ f ( ph  /\  (
v  e.  ( ( A  Func  C )  X.  ( A  Func  C
) )  /\  h  e.  ( A  Func  C
) ) )
16 nfcsb1v 3446 . . . . . . 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 11 . . . . . 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 5882 . . . . . . 7  |-  ( 1st `  v )  e.  _V
1918a1i 11 . . . . . 6  |-  ( (
ph  /\  ( v  e.  ( ( A  Func  C )  X.  ( A 
Func  C ) )  /\  h  e.  ( A  Func  C ) ) )  ->  ( 1st `  v
)  e.  _V )
20 nfv 1708 . . . . . . . 8  |-  F/ g ( ( ph  /\  ( v  e.  ( ( A  Func  C
)  X.  ( A 
Func  C ) )  /\  h  e.  ( A  Func  C ) ) )  /\  f  =  ( 1st `  v ) )
21 nfcsb1v 3446 . . . . . . . . 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 11 . . . . . . . 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 5882 . . . . . . . . 9  |-  ( 2nd `  v )  e.  _V
2423a1i 11 . . . . . . . 8  |-  ( ( ( ph  /\  (
v  e.  ( ( A  Func  C )  X.  ( A  Func  C
) )  /\  h  e.  ( A  Func  C
) ) )  /\  f  =  ( 1st `  v ) )  -> 
( 2nd `  v
)  e.  _V )
2511ad3antrrr 729 . . . . . . . . . . 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 6313 . . . . . . . . . 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 ) )
2725oveqdr 6320 . . . . . . . . . 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 ) )
281homfeqbas 15112 . . . . . . . . . . . 12  |-  ( ph  ->  ( Base `  A
)  =  ( Base `  B ) )
2928ad4antr 731 . . . . . . . . . . 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
) )
30 eqid 2457 . . . . . . . . . . . 12  |-  ( Base `  C )  =  (
Base `  C )
31 eqid 2457 . . . . . . . . . . . 12  |-  ( Hom  `  C )  =  ( Hom  `  C )
32 eqid 2457 . . . . . . . . . . . 12  |-  (comp `  C )  =  (comp `  C )
33 eqid 2457 . . . . . . . . . . . 12  |-  (comp `  D )  =  (comp `  D )
343ad5antr 733 . . . . . . . . . . . 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
) )  ->  ( Hom f  `  C )  =  ( Hom f  `  D ) )
354ad5antr 733 . . . . . . . . . . . 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
) )
36 eqid 2457 . . . . . . . . . . . . . 14  |-  ( Base `  A )  =  (
Base `  A )
37 relfunc 15278 . . . . . . . . . . . . . . 15  |-  Rel  ( A  Func  C )
38 simpllr 760 . . . . . . . . . . . . . . . 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
) )
39 simp-4r 768 . . . . . . . . . . . . . . . . . 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
) ) )
4039simpld 459 . . . . . . . . . . . . . . . . 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 ) ) )
41 xp1st 6829 . . . . . . . . . . . . . . . . 17  |-  ( v  e.  ( ( A 
Func  C )  X.  ( A  Func  C ) )  ->  ( 1st `  v
)  e.  ( A 
Func  C ) )
4240, 41syl 16 . . . . . . . . . . . . . . . 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
) )
4338, 42eqeltrd 2545 . . . . . . . . . . . . . . 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
) )
44 1st2ndbr 6848 . . . . . . . . . . . . . . 15  |-  ( ( Rel  ( A  Func  C )  /\  f  e.  ( A  Func  C
) )  ->  ( 1st `  f ) ( A  Func  C )
( 2nd `  f
) )
4537, 43, 44sylancr 663 . . . . . . . . . . . . . 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
) )
4636, 30, 45funcf1 15282 . . . . . . . . . . . . 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
) )
4746ffvelrnda 6032 . . . . . . . . . . . 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
) )
48 simplr 755 . . . . . . . . . . . . . . . 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
) )
49 xp2nd 6830 . . . . . . . . . . . . . . . . 17  |-  ( v  e.  ( ( A 
Func  C )  X.  ( A  Func  C ) )  ->  ( 2nd `  v
)  e.  ( A 
Func  C ) )
5040, 49syl 16 . . . . . . . . . . . . . . . 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
) )
5148, 50eqeltrd 2545 . . . . . . . . . . . . . . 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
) )
52 1st2ndbr 6848 . . . . . . . . . . . . . . 15  |-  ( ( Rel  ( A  Func  C )  /\  g  e.  ( A  Func  C
) )  ->  ( 1st `  g ) ( A  Func  C )
( 2nd `  g
) )
5337, 51, 52sylancr 663 . . . . . . . . . . . . . 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
) )
5436, 30, 53funcf1 15282 . . . . . . . . . . . . 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
) )
5554ffvelrnda 6032 . . . . . . . . . . . 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
) )
5639simprd 463 . . . . . . . . . . . . . . 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
) )
57 1st2ndbr 6848 . . . . . . . . . . . . . . 15  |-  ( ( Rel  ( A  Func  C )  /\  h  e.  ( A  Func  C
) )  ->  ( 1st `  h ) ( A  Func  C )
( 2nd `  h
) )
5837, 56, 57sylancr 663 . . . . . . . . . . . . . 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
) )
5936, 30, 58funcf1 15282 . . . . . . . . . . . . 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
) )
6059ffvelrnda 6032 . . . . . . . . . . . 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
) )
61 eqid 2457 . . . . . . . . . . . . 13  |-  ( A Nat 
C )  =  ( A Nat  C )
62 simplrr 762 . . . . . . . . . . . . . 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 ) )
6361, 62nat1st2nd 15367 . . . . . . . . . . . . 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
) >. ) )
64 simpr 461 . . . . . . . . . . . . 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
) )
6561, 63, 36, 31, 64natcl 15369 . . . . . . . . . . . 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
) ) )
66 simplrl 761 . . . . . . . . . . . . . 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 ) )
6761, 66nat1st2nd 15367 . . . . . . . . . . . . 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
) >. ) )
6861, 67, 36, 31, 64natcl 15369 . . . . . . . . . . . 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
) ) )
6930, 31, 32, 33, 34, 35, 47, 55, 60, 65, 68comfeqval 15124 . . . . . . . . . . 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 ) ) )
7029, 69mpteq12dva 4534 . . . . . . . . . 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 ) ) ) )
7126, 27, 70mpt2eq123dva 6357 . . . . . . . . 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 ) ) ) ) )
72 csbeq1a 3439 . . . . . . . . . 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 ) ) ) ) )
7372adantl 466 . . . . . . . . 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 ) ) ) ) )
7471, 73eqtrd 2498 . . . . . . . 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 ) ) ) ) )
7520, 22, 24, 74csbiedf 3451 . . . . . . 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 ) ) ) ) )
76 csbeq1a 3439 . . . . . . . 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 ) ) ) ) )
7776adantl 466 . . . . . . 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 ) ) ) ) )
7875, 77eqtrd 2498 . . . . . 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 ) ) ) ) )
7915, 17, 19, 78csbiedf 3451 . . . . 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 ) ) ) ) )
8013, 14, 79mpt2eq123dva 6357 . . . 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 ) ) ) ) ) )
8180opeq2d 4226 . . 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 ) ) ) ) ) >.
)
8210, 12, 81tpeq123d 4126 . 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 ) ) ) ) ) >. } )
83 eqid 2457 . . 3  |-  ( A FuncCat  C )  =  ( A FuncCat  C )
84 eqid 2457 . . 3  |-  ( A 
Func  C )  =  ( A  Func  C )
85 eqidd 2458 . . 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 ) ) ) ) ) )
8683, 84, 61, 36, 32, 5, 7, 85fucval 15374 . 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 ) ) ) ) ) >. } )
87 eqid 2457 . . 3  |-  ( B FuncCat  D )  =  ( B FuncCat  D )
88 eqid 2457 . . 3  |-  ( B 
Func  D )  =  ( B  Func  D )
89 eqid 2457 . . 3  |-  ( B Nat 
D )  =  ( B Nat  D )
90 eqid 2457 . . 3  |-  ( Base `  B )  =  (
Base `  B )
91 eqidd 2458 . . 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 ) ) ) ) ) )
9287, 88, 89, 90, 33, 6, 8, 91fucval 15374 . 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 ) ) ) ) ) >. } )
9382, 86, 923eqtr4d 2508 1  |-  ( ph  ->  ( A FuncCat  C )  =  ( B FuncCat  D
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1395    e. wcel 1819   F/_wnfc 2605   _Vcvv 3109   [_csb 3430   {ctp 4036   <.cop 4038   class class class wbr 4456    |-> cmpt 4515    X. cxp 5006   Rel wrel 5013   ` cfv 5594  (class class class)co 6296    |-> cmpt2 6298   1stc1st 6797   2ndc2nd 6798   ndxcnx 14641   Basecbs 14644   Hom chom 14723  compcco 14724   Catccat 15081   Hom f chomf 15083  compfccomf 15084    Func cfunc 15270   Nat cnat 15357   FuncCat cfuc 15358
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-fal 1401  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6799  df-2nd 6800  df-map 7440  df-ixp 7489  df-cat 15085  df-cid 15086  df-homf 15087  df-comf 15088  df-func 15274  df-nat 15359  df-fuc 15360
This theorem is referenced by:  oyoncl  15666
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