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Theorem xpcpropd 14031
Description: If two categories have the same set of objects, morphisms, and compositions, then they have the same product category. (Contributed by Mario Carneiro, 17-Jan-2017.)
Hypotheses
Ref Expression
xpcpropd.1  |-  ( ph  ->  (  Homf 
`  A )  =  (  Homf 
`  B ) )
xpcpropd.2  |-  ( ph  ->  (compf `  A )  =  (compf `  B ) )
xpcpropd.3  |-  ( ph  ->  (  Homf 
`  C )  =  (  Homf 
`  D ) )
xpcpropd.4  |-  ( ph  ->  (compf `  C )  =  (compf `  D ) )
xpcpropd.a  |-  ( ph  ->  A  e.  V )
xpcpropd.b  |-  ( ph  ->  B  e.  V )
xpcpropd.c  |-  ( ph  ->  C  e.  V )
xpcpropd.d  |-  ( ph  ->  D  e.  V )
Assertion
Ref Expression
xpcpropd  |-  ( ph  ->  ( A  X.c  C )  =  ( B  X.c  D
) )

Proof of Theorem xpcpropd
Dummy variables  f 
g  u  v  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2316 . . 3  |-  ( A  X.c  C )  =  ( A  X.c  C )
2 eqid 2316 . . 3  |-  ( Base `  A )  =  (
Base `  A )
3 eqid 2316 . . 3  |-  ( Base `  C )  =  (
Base `  C )
4 eqid 2316 . . 3  |-  (  Hom  `  A )  =  (  Hom  `  A )
5 eqid 2316 . . 3  |-  (  Hom  `  C )  =  (  Hom  `  C )
6 eqid 2316 . . 3  |-  (comp `  A )  =  (comp `  A )
7 eqid 2316 . . 3  |-  (comp `  C )  =  (comp `  C )
8 xpcpropd.a . . 3  |-  ( ph  ->  A  e.  V )
9 xpcpropd.c . . 3  |-  ( ph  ->  C  e.  V )
10 eqidd 2317 . . 3  |-  ( ph  ->  ( ( Base `  A
)  X.  ( Base `  C ) )  =  ( ( Base `  A
)  X.  ( Base `  C ) ) )
111, 2, 3xpcbas 14001 . . . . 5  |-  ( (
Base `  A )  X.  ( Base `  C
) )  =  (
Base `  ( A  X.c  C ) )
12 eqid 2316 . . . . 5  |-  (  Hom  `  ( A  X.c  C ) )  =  (  Hom  `  ( A  X.c  C ) )
131, 11, 4, 5, 12xpchomfval 14002 . . . 4  |-  (  Hom  `  ( A  X.c  C ) )  =  ( u  e.  ( ( Base `  A )  X.  ( Base `  C ) ) ,  v  e.  ( ( Base `  A
)  X.  ( Base `  C ) )  |->  ( ( ( 1st `  u
) (  Hom  `  A
) ( 1st `  v
) )  X.  (
( 2nd `  u
) (  Hom  `  C
) ( 2nd `  v
) ) ) )
1413a1i 10 . . 3  |-  ( ph  ->  (  Hom  `  ( A  X.c  C ) )  =  ( u  e.  ( ( Base `  A
)  X.  ( Base `  C ) ) ,  v  e.  ( (
Base `  A )  X.  ( Base `  C
) )  |->  ( ( ( 1st `  u
) (  Hom  `  A
) ( 1st `  v
) )  X.  (
( 2nd `  u
) (  Hom  `  C
) ( 2nd `  v
) ) ) ) )
15 eqidd 2317 . . 3  |-  ( ph  ->  ( x  e.  ( ( ( Base `  A
)  X.  ( Base `  C ) )  X.  ( ( Base `  A
)  X.  ( Base `  C ) ) ) ,  y  e.  ( ( Base `  A
)  X.  ( Base `  C ) )  |->  ( g  e.  ( ( 2nd `  x ) (  Hom  `  ( A  X.c  C ) ) y ) ,  f  e.  ( (  Hom  `  ( A  X.c  C ) ) `  x )  |->  <. (
( 1st `  g
) ( <. ( 1st `  ( 1st `  x
) ) ,  ( 1st `  ( 2nd `  x ) ) >.
(comp `  A )
( 1st `  y
) ) ( 1st `  f ) ) ,  ( ( 2nd `  g
) ( <. ( 2nd `  ( 1st `  x
) ) ,  ( 2nd `  ( 2nd `  x ) ) >.
(comp `  C )
( 2nd `  y
) ) ( 2nd `  f ) ) >.
) )  =  ( x  e.  ( ( ( Base `  A
)  X.  ( Base `  C ) )  X.  ( ( Base `  A
)  X.  ( Base `  C ) ) ) ,  y  e.  ( ( Base `  A
)  X.  ( Base `  C ) )  |->  ( g  e.  ( ( 2nd `  x ) (  Hom  `  ( A  X.c  C ) ) y ) ,  f  e.  ( (  Hom  `  ( A  X.c  C ) ) `  x )  |->  <. (
( 1st `  g
) ( <. ( 1st `  ( 1st `  x
) ) ,  ( 1st `  ( 2nd `  x ) ) >.
(comp `  A )
( 1st `  y
) ) ( 1st `  f ) ) ,  ( ( 2nd `  g
) ( <. ( 2nd `  ( 1st `  x
) ) ,  ( 2nd `  ( 2nd `  x ) ) >.
(comp `  C )
( 2nd `  y
) ) ( 2nd `  f ) ) >.
) ) )
161, 2, 3, 4, 5, 6, 7, 8, 9, 10, 14, 15xpcval 14000 . 2  |-  ( ph  ->  ( A  X.c  C )  =  { <. ( Base `  ndx ) ,  ( ( Base `  A
)  X.  ( Base `  C ) ) >. ,  <. (  Hom  `  ndx ) ,  (  Hom  `  ( A  X.c  C ) ) >. ,  <. (comp ` 
ndx ) ,  ( x  e.  ( ( ( Base `  A
)  X.  ( Base `  C ) )  X.  ( ( Base `  A
)  X.  ( Base `  C ) ) ) ,  y  e.  ( ( Base `  A
)  X.  ( Base `  C ) )  |->  ( g  e.  ( ( 2nd `  x ) (  Hom  `  ( A  X.c  C ) ) y ) ,  f  e.  ( (  Hom  `  ( A  X.c  C ) ) `  x )  |->  <. (
( 1st `  g
) ( <. ( 1st `  ( 1st `  x
) ) ,  ( 1st `  ( 2nd `  x ) ) >.
(comp `  A )
( 1st `  y
) ) ( 1st `  f ) ) ,  ( ( 2nd `  g
) ( <. ( 2nd `  ( 1st `  x
) ) ,  ( 2nd `  ( 2nd `  x ) ) >.
(comp `  C )
( 2nd `  y
) ) ( 2nd `  f ) ) >.
) ) >. } )
17 eqid 2316 . . 3  |-  ( B  X.c  D )  =  ( B  X.c  D )
18 eqid 2316 . . 3  |-  ( Base `  B )  =  (
Base `  B )
19 eqid 2316 . . 3  |-  ( Base `  D )  =  (
Base `  D )
20 eqid 2316 . . 3  |-  (  Hom  `  B )  =  (  Hom  `  B )
21 eqid 2316 . . 3  |-  (  Hom  `  D )  =  (  Hom  `  D )
22 eqid 2316 . . 3  |-  (comp `  B )  =  (comp `  B )
23 eqid 2316 . . 3  |-  (comp `  D )  =  (comp `  D )
24 xpcpropd.b . . 3  |-  ( ph  ->  B  e.  V )
25 xpcpropd.d . . 3  |-  ( ph  ->  D  e.  V )
26 xpcpropd.1 . . . . 5  |-  ( ph  ->  (  Homf 
`  A )  =  (  Homf 
`  B ) )
2726homfeqbas 13648 . . . 4  |-  ( ph  ->  ( Base `  A
)  =  ( Base `  B ) )
28 xpcpropd.3 . . . . 5  |-  ( ph  ->  (  Homf 
`  C )  =  (  Homf 
`  D ) )
2928homfeqbas 13648 . . . 4  |-  ( ph  ->  ( Base `  C
)  =  ( Base `  D ) )
3027, 29xpeq12d 4751 . . 3  |-  ( ph  ->  ( ( Base `  A
)  X.  ( Base `  C ) )  =  ( ( Base `  B
)  X.  ( Base `  D ) ) )
31263ad2ant1 976 . . . . . . 7  |-  ( (
ph  /\  u  e.  ( ( Base `  A
)  X.  ( Base `  C ) )  /\  v  e.  ( ( Base `  A )  X.  ( Base `  C
) ) )  -> 
(  Homf 
`  A )  =  (  Homf 
`  B ) )
32 xp1st 6191 . . . . . . . 8  |-  ( u  e.  ( ( Base `  A )  X.  ( Base `  C ) )  ->  ( 1st `  u
)  e.  ( Base `  A ) )
33323ad2ant2 977 . . . . . . 7  |-  ( (
ph  /\  u  e.  ( ( Base `  A
)  X.  ( Base `  C ) )  /\  v  e.  ( ( Base `  A )  X.  ( Base `  C
) ) )  -> 
( 1st `  u
)  e.  ( Base `  A ) )
34 xp1st 6191 . . . . . . . 8  |-  ( v  e.  ( ( Base `  A )  X.  ( Base `  C ) )  ->  ( 1st `  v
)  e.  ( Base `  A ) )
35343ad2ant3 978 . . . . . . 7  |-  ( (
ph  /\  u  e.  ( ( Base `  A
)  X.  ( Base `  C ) )  /\  v  e.  ( ( Base `  A )  X.  ( Base `  C
) ) )  -> 
( 1st `  v
)  e.  ( Base `  A ) )
362, 4, 20, 31, 33, 35homfeqval 13649 . . . . . 6  |-  ( (
ph  /\  u  e.  ( ( Base `  A
)  X.  ( Base `  C ) )  /\  v  e.  ( ( Base `  A )  X.  ( Base `  C
) ) )  -> 
( ( 1st `  u
) (  Hom  `  A
) ( 1st `  v
) )  =  ( ( 1st `  u
) (  Hom  `  B
) ( 1st `  v
) ) )
37283ad2ant1 976 . . . . . . 7  |-  ( (
ph  /\  u  e.  ( ( Base `  A
)  X.  ( Base `  C ) )  /\  v  e.  ( ( Base `  A )  X.  ( Base `  C
) ) )  -> 
(  Homf 
`  C )  =  (  Homf 
`  D ) )
38 xp2nd 6192 . . . . . . . 8  |-  ( u  e.  ( ( Base `  A )  X.  ( Base `  C ) )  ->  ( 2nd `  u
)  e.  ( Base `  C ) )
39383ad2ant2 977 . . . . . . 7  |-  ( (
ph  /\  u  e.  ( ( Base `  A
)  X.  ( Base `  C ) )  /\  v  e.  ( ( Base `  A )  X.  ( Base `  C
) ) )  -> 
( 2nd `  u
)  e.  ( Base `  C ) )
40 xp2nd 6192 . . . . . . . 8  |-  ( v  e.  ( ( Base `  A )  X.  ( Base `  C ) )  ->  ( 2nd `  v
)  e.  ( Base `  C ) )
41403ad2ant3 978 . . . . . . 7  |-  ( (
ph  /\  u  e.  ( ( Base `  A
)  X.  ( Base `  C ) )  /\  v  e.  ( ( Base `  A )  X.  ( Base `  C
) ) )  -> 
( 2nd `  v
)  e.  ( Base `  C ) )
423, 5, 21, 37, 39, 41homfeqval 13649 . . . . . 6  |-  ( (
ph  /\  u  e.  ( ( Base `  A
)  X.  ( Base `  C ) )  /\  v  e.  ( ( Base `  A )  X.  ( Base `  C
) ) )  -> 
( ( 2nd `  u
) (  Hom  `  C
) ( 2nd `  v
) )  =  ( ( 2nd `  u
) (  Hom  `  D
) ( 2nd `  v
) ) )
4336, 42xpeq12d 4751 . . . . 5  |-  ( (
ph  /\  u  e.  ( ( Base `  A
)  X.  ( Base `  C ) )  /\  v  e.  ( ( Base `  A )  X.  ( Base `  C
) ) )  -> 
( ( ( 1st `  u ) (  Hom  `  A ) ( 1st `  v ) )  X.  ( ( 2nd `  u
) (  Hom  `  C
) ( 2nd `  v
) ) )  =  ( ( ( 1st `  u ) (  Hom  `  B ) ( 1st `  v ) )  X.  ( ( 2nd `  u
) (  Hom  `  D
) ( 2nd `  v
) ) ) )
4443mpt2eq3dva 5954 . . . 4  |-  ( ph  ->  ( u  e.  ( ( Base `  A
)  X.  ( Base `  C ) ) ,  v  e.  ( (
Base `  A )  X.  ( Base `  C
) )  |->  ( ( ( 1st `  u
) (  Hom  `  A
) ( 1st `  v
) )  X.  (
( 2nd `  u
) (  Hom  `  C
) ( 2nd `  v
) ) ) )  =  ( u  e.  ( ( Base `  A
)  X.  ( Base `  C ) ) ,  v  e.  ( (
Base `  A )  X.  ( Base `  C
) )  |->  ( ( ( 1st `  u
) (  Hom  `  B
) ( 1st `  v
) )  X.  (
( 2nd `  u
) (  Hom  `  D
) ( 2nd `  v
) ) ) ) )
4513, 44syl5eq 2360 . . 3  |-  ( ph  ->  (  Hom  `  ( A  X.c  C ) )  =  ( u  e.  ( ( Base `  A
)  X.  ( Base `  C ) ) ,  v  e.  ( (
Base `  A )  X.  ( Base `  C
) )  |->  ( ( ( 1st `  u
) (  Hom  `  B
) ( 1st `  v
) )  X.  (
( 2nd `  u
) (  Hom  `  D
) ( 2nd `  v
) ) ) ) )
4626ad4antr 712 . . . . . . . . 9  |-  ( ( ( ( ( ph  /\  x  e.  ( ( ( Base `  A
)  X.  ( Base `  C ) )  X.  ( ( Base `  A
)  X.  ( Base `  C ) ) ) )  /\  y  e.  ( ( Base `  A
)  X.  ( Base `  C ) ) )  /\  g  e.  ( ( 2nd `  x
) (  Hom  `  ( A  X.c  C ) ) y ) )  /\  f  e.  ( (  Hom  `  ( A  X.c  C ) ) `  x ) )  -> 
(  Homf 
`  A )  =  (  Homf 
`  B ) )
47 xpcpropd.2 . . . . . . . . . 10  |-  ( ph  ->  (compf `  A )  =  (compf `  B ) )
4847ad4antr 712 . . . . . . . . 9  |-  ( ( ( ( ( ph  /\  x  e.  ( ( ( Base `  A
)  X.  ( Base `  C ) )  X.  ( ( Base `  A
)  X.  ( Base `  C ) ) ) )  /\  y  e.  ( ( Base `  A
)  X.  ( Base `  C ) ) )  /\  g  e.  ( ( 2nd `  x
) (  Hom  `  ( A  X.c  C ) ) y ) )  /\  f  e.  ( (  Hom  `  ( A  X.c  C ) ) `  x ) )  -> 
(compf `  A )  =  (compf `  B ) )
49 simp-4r 743 . . . . . . . . . . 11  |-  ( ( ( ( ( ph  /\  x  e.  ( ( ( Base `  A
)  X.  ( Base `  C ) )  X.  ( ( Base `  A
)  X.  ( Base `  C ) ) ) )  /\  y  e.  ( ( Base `  A
)  X.  ( Base `  C ) ) )  /\  g  e.  ( ( 2nd `  x
) (  Hom  `  ( A  X.c  C ) ) y ) )  /\  f  e.  ( (  Hom  `  ( A  X.c  C ) ) `  x ) )  ->  x  e.  ( (
( Base `  A )  X.  ( Base `  C
) )  X.  (
( Base `  A )  X.  ( Base `  C
) ) ) )
50 xp1st 6191 . . . . . . . . . . 11  |-  ( x  e.  ( ( (
Base `  A )  X.  ( Base `  C
) )  X.  (
( Base `  A )  X.  ( Base `  C
) ) )  -> 
( 1st `  x
)  e.  ( (
Base `  A )  X.  ( Base `  C
) ) )
5149, 50syl 15 . . . . . . . . . 10  |-  ( ( ( ( ( ph  /\  x  e.  ( ( ( Base `  A
)  X.  ( Base `  C ) )  X.  ( ( Base `  A
)  X.  ( Base `  C ) ) ) )  /\  y  e.  ( ( Base `  A
)  X.  ( Base `  C ) ) )  /\  g  e.  ( ( 2nd `  x
) (  Hom  `  ( A  X.c  C ) ) y ) )  /\  f  e.  ( (  Hom  `  ( A  X.c  C ) ) `  x ) )  -> 
( 1st `  x
)  e.  ( (
Base `  A )  X.  ( Base `  C
) ) )
52 xp1st 6191 . . . . . . . . . 10  |-  ( ( 1st `  x )  e.  ( ( Base `  A )  X.  ( Base `  C ) )  ->  ( 1st `  ( 1st `  x ) )  e.  ( Base `  A
) )
5351, 52syl 15 . . . . . . . . 9  |-  ( ( ( ( ( ph  /\  x  e.  ( ( ( Base `  A
)  X.  ( Base `  C ) )  X.  ( ( Base `  A
)  X.  ( Base `  C ) ) ) )  /\  y  e.  ( ( Base `  A
)  X.  ( Base `  C ) ) )  /\  g  e.  ( ( 2nd `  x
) (  Hom  `  ( A  X.c  C ) ) y ) )  /\  f  e.  ( (  Hom  `  ( A  X.c  C ) ) `  x ) )  -> 
( 1st `  ( 1st `  x ) )  e.  ( Base `  A
) )
54 xp2nd 6192 . . . . . . . . . . 11  |-  ( x  e.  ( ( (
Base `  A )  X.  ( Base `  C
) )  X.  (
( Base `  A )  X.  ( Base `  C
) ) )  -> 
( 2nd `  x
)  e.  ( (
Base `  A )  X.  ( Base `  C
) ) )
5549, 54syl 15 . . . . . . . . . 10  |-  ( ( ( ( ( ph  /\  x  e.  ( ( ( Base `  A
)  X.  ( Base `  C ) )  X.  ( ( Base `  A
)  X.  ( Base `  C ) ) ) )  /\  y  e.  ( ( Base `  A
)  X.  ( Base `  C ) ) )  /\  g  e.  ( ( 2nd `  x
) (  Hom  `  ( A  X.c  C ) ) y ) )  /\  f  e.  ( (  Hom  `  ( A  X.c  C ) ) `  x ) )  -> 
( 2nd `  x
)  e.  ( (
Base `  A )  X.  ( Base `  C
) ) )
56 xp1st 6191 . . . . . . . . . 10  |-  ( ( 2nd `  x )  e.  ( ( Base `  A )  X.  ( Base `  C ) )  ->  ( 1st `  ( 2nd `  x ) )  e.  ( Base `  A
) )
5755, 56syl 15 . . . . . . . . 9  |-  ( ( ( ( ( ph  /\  x  e.  ( ( ( Base `  A
)  X.  ( Base `  C ) )  X.  ( ( Base `  A
)  X.  ( Base `  C ) ) ) )  /\  y  e.  ( ( Base `  A
)  X.  ( Base `  C ) ) )  /\  g  e.  ( ( 2nd `  x
) (  Hom  `  ( A  X.c  C ) ) y ) )  /\  f  e.  ( (  Hom  `  ( A  X.c  C ) ) `  x ) )  -> 
( 1st `  ( 2nd `  x ) )  e.  ( Base `  A
) )
58 simpllr 735 . . . . . . . . . 10  |-  ( ( ( ( ( ph  /\  x  e.  ( ( ( Base `  A
)  X.  ( Base `  C ) )  X.  ( ( Base `  A
)  X.  ( Base `  C ) ) ) )  /\  y  e.  ( ( Base `  A
)  X.  ( Base `  C ) ) )  /\  g  e.  ( ( 2nd `  x
) (  Hom  `  ( A  X.c  C ) ) y ) )  /\  f  e.  ( (  Hom  `  ( A  X.c  C ) ) `  x ) )  -> 
y  e.  ( (
Base `  A )  X.  ( Base `  C
) ) )
59 xp1st 6191 . . . . . . . . . 10  |-  ( y  e.  ( ( Base `  A )  X.  ( Base `  C ) )  ->  ( 1st `  y
)  e.  ( Base `  A ) )
6058, 59syl 15 . . . . . . . . 9  |-  ( ( ( ( ( ph  /\  x  e.  ( ( ( Base `  A
)  X.  ( Base `  C ) )  X.  ( ( Base `  A
)  X.  ( Base `  C ) ) ) )  /\  y  e.  ( ( Base `  A
)  X.  ( Base `  C ) ) )  /\  g  e.  ( ( 2nd `  x
) (  Hom  `  ( A  X.c  C ) ) y ) )  /\  f  e.  ( (  Hom  `  ( A  X.c  C ) ) `  x ) )  -> 
( 1st `  y
)  e.  ( Base `  A ) )
61 simpr 447 . . . . . . . . . . 11  |-  ( ( ( ( ( ph  /\  x  e.  ( ( ( Base `  A
)  X.  ( Base `  C ) )  X.  ( ( Base `  A
)  X.  ( Base `  C ) ) ) )  /\  y  e.  ( ( Base `  A
)  X.  ( Base `  C ) ) )  /\  g  e.  ( ( 2nd `  x
) (  Hom  `  ( A  X.c  C ) ) y ) )  /\  f  e.  ( (  Hom  `  ( A  X.c  C ) ) `  x ) )  -> 
f  e.  ( (  Hom  `  ( A  X.c  C ) ) `  x ) )
62 1st2nd2 6201 . . . . . . . . . . . . . . 15  |-  ( x  e.  ( ( (
Base `  A )  X.  ( Base `  C
) )  X.  (
( Base `  A )  X.  ( Base `  C
) ) )  ->  x  =  <. ( 1st `  x ) ,  ( 2nd `  x )
>. )
6349, 62syl 15 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ph  /\  x  e.  ( ( ( Base `  A
)  X.  ( Base `  C ) )  X.  ( ( Base `  A
)  X.  ( Base `  C ) ) ) )  /\  y  e.  ( ( Base `  A
)  X.  ( Base `  C ) ) )  /\  g  e.  ( ( 2nd `  x
) (  Hom  `  ( A  X.c  C ) ) y ) )  /\  f  e.  ( (  Hom  `  ( A  X.c  C ) ) `  x ) )  ->  x  =  <. ( 1st `  x ) ,  ( 2nd `  x )
>. )
6463fveq2d 5567 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ph  /\  x  e.  ( ( ( Base `  A
)  X.  ( Base `  C ) )  X.  ( ( Base `  A
)  X.  ( Base `  C ) ) ) )  /\  y  e.  ( ( Base `  A
)  X.  ( Base `  C ) ) )  /\  g  e.  ( ( 2nd `  x
) (  Hom  `  ( A  X.c  C ) ) y ) )  /\  f  e.  ( (  Hom  `  ( A  X.c  C ) ) `  x ) )  -> 
( (  Hom  `  ( A  X.c  C ) ) `  x )  =  ( (  Hom  `  ( A  X.c  C ) ) `  <. ( 1st `  x
) ,  ( 2nd `  x ) >. )
)
65 df-ov 5903 . . . . . . . . . . . . 13  |-  ( ( 1st `  x ) (  Hom  `  ( A  X.c  C ) ) ( 2nd `  x ) )  =  ( (  Hom  `  ( A  X.c  C ) ) `  <. ( 1st `  x
) ,  ( 2nd `  x ) >. )
6664, 65syl6eqr 2366 . . . . . . . . . . . 12  |-  ( ( ( ( ( ph  /\  x  e.  ( ( ( Base `  A
)  X.  ( Base `  C ) )  X.  ( ( Base `  A
)  X.  ( Base `  C ) ) ) )  /\  y  e.  ( ( Base `  A
)  X.  ( Base `  C ) ) )  /\  g  e.  ( ( 2nd `  x
) (  Hom  `  ( A  X.c  C ) ) y ) )  /\  f  e.  ( (  Hom  `  ( A  X.c  C ) ) `  x ) )  -> 
( (  Hom  `  ( A  X.c  C ) ) `  x )  =  ( ( 1st `  x
) (  Hom  `  ( A  X.c  C ) ) ( 2nd `  x ) ) )
671, 11, 4, 5, 12, 51, 55xpchom 14003 . . . . . . . . . . . 12  |-  ( ( ( ( ( ph  /\  x  e.  ( ( ( Base `  A
)  X.  ( Base `  C ) )  X.  ( ( Base `  A
)  X.  ( Base `  C ) ) ) )  /\  y  e.  ( ( Base `  A
)  X.  ( Base `  C ) ) )  /\  g  e.  ( ( 2nd `  x
) (  Hom  `  ( A  X.c  C ) ) y ) )  /\  f  e.  ( (  Hom  `  ( A  X.c  C ) ) `  x ) )  -> 
( ( 1st `  x
) (  Hom  `  ( A  X.c  C ) ) ( 2nd `  x ) )  =  ( ( ( 1st `  ( 1st `  x ) ) (  Hom  `  A
) ( 1st `  ( 2nd `  x ) ) )  X.  ( ( 2nd `  ( 1st `  x ) ) (  Hom  `  C )
( 2nd `  ( 2nd `  x ) ) ) ) )
6866, 67eqtrd 2348 . . . . . . . . . . 11  |-  ( ( ( ( ( ph  /\  x  e.  ( ( ( Base `  A
)  X.  ( Base `  C ) )  X.  ( ( Base `  A
)  X.  ( Base `  C ) ) ) )  /\  y  e.  ( ( Base `  A
)  X.  ( Base `  C ) ) )  /\  g  e.  ( ( 2nd `  x
) (  Hom  `  ( A  X.c  C ) ) y ) )  /\  f  e.  ( (  Hom  `  ( A  X.c  C ) ) `  x ) )  -> 
( (  Hom  `  ( A  X.c  C ) ) `  x )  =  ( ( ( 1st `  ( 1st `  x ) ) (  Hom  `  A
) ( 1st `  ( 2nd `  x ) ) )  X.  ( ( 2nd `  ( 1st `  x ) ) (  Hom  `  C )
( 2nd `  ( 2nd `  x ) ) ) ) )
6961, 68eleqtrd 2392 . . . . . . . . . 10  |-  ( ( ( ( ( ph  /\  x  e.  ( ( ( Base `  A
)  X.  ( Base `  C ) )  X.  ( ( Base `  A
)  X.  ( Base `  C ) ) ) )  /\  y  e.  ( ( Base `  A
)  X.  ( Base `  C ) ) )  /\  g  e.  ( ( 2nd `  x
) (  Hom  `  ( A  X.c  C ) ) y ) )  /\  f  e.  ( (  Hom  `  ( A  X.c  C ) ) `  x ) )  -> 
f  e.  ( ( ( 1st `  ( 1st `  x ) ) (  Hom  `  A
) ( 1st `  ( 2nd `  x ) ) )  X.  ( ( 2nd `  ( 1st `  x ) ) (  Hom  `  C )
( 2nd `  ( 2nd `  x ) ) ) ) )
70 xp1st 6191 . . . . . . . . . 10  |-  ( f  e.  ( ( ( 1st `  ( 1st `  x ) ) (  Hom  `  A )
( 1st `  ( 2nd `  x ) ) )  X.  ( ( 2nd `  ( 1st `  x ) ) (  Hom  `  C )
( 2nd `  ( 2nd `  x ) ) ) )  ->  ( 1st `  f )  e.  ( ( 1st `  ( 1st `  x ) ) (  Hom  `  A
) ( 1st `  ( 2nd `  x ) ) ) )
7169, 70syl 15 . . . . . . . . 9  |-  ( ( ( ( ( ph  /\  x  e.  ( ( ( Base `  A
)  X.  ( Base `  C ) )  X.  ( ( Base `  A
)  X.  ( Base `  C ) ) ) )  /\  y  e.  ( ( Base `  A
)  X.  ( Base `  C ) ) )  /\  g  e.  ( ( 2nd `  x
) (  Hom  `  ( A  X.c  C ) ) y ) )  /\  f  e.  ( (  Hom  `  ( A  X.c  C ) ) `  x ) )  -> 
( 1st `  f
)  e.  ( ( 1st `  ( 1st `  x ) ) (  Hom  `  A )
( 1st `  ( 2nd `  x ) ) ) )
72 simplr 731 . . . . . . . . . . 11  |-  ( ( ( ( ( ph  /\  x  e.  ( ( ( Base `  A
)  X.  ( Base `  C ) )  X.  ( ( Base `  A
)  X.  ( Base `  C ) ) ) )  /\  y  e.  ( ( Base `  A
)  X.  ( Base `  C ) ) )  /\  g  e.  ( ( 2nd `  x
) (  Hom  `  ( A  X.c  C ) ) y ) )  /\  f  e.  ( (  Hom  `  ( A  X.c  C ) ) `  x ) )  -> 
g  e.  ( ( 2nd `  x ) (  Hom  `  ( A  X.c  C ) ) y ) )
731, 11, 4, 5, 12, 55, 58xpchom 14003 . . . . . . . . . . 11  |-  ( ( ( ( ( ph  /\  x  e.  ( ( ( Base `  A
)  X.  ( Base `  C ) )  X.  ( ( Base `  A
)  X.  ( Base `  C ) ) ) )  /\  y  e.  ( ( Base `  A
)  X.  ( Base `  C ) ) )  /\  g  e.  ( ( 2nd `  x
) (  Hom  `  ( A  X.c  C ) ) y ) )  /\  f  e.  ( (  Hom  `  ( A  X.c  C ) ) `  x ) )  -> 
( ( 2nd `  x
) (  Hom  `  ( A  X.c  C ) ) y )  =  ( ( ( 1st `  ( 2nd `  x ) ) (  Hom  `  A
) ( 1st `  y
) )  X.  (
( 2nd `  ( 2nd `  x ) ) (  Hom  `  C
) ( 2nd `  y
) ) ) )
7472, 73eleqtrd 2392 . . . . . . . . . 10  |-  ( ( ( ( ( ph  /\  x  e.  ( ( ( Base `  A
)  X.  ( Base `  C ) )  X.  ( ( Base `  A
)  X.  ( Base `  C ) ) ) )  /\  y  e.  ( ( Base `  A
)  X.  ( Base `  C ) ) )  /\  g  e.  ( ( 2nd `  x
) (  Hom  `  ( A  X.c  C ) ) y ) )  /\  f  e.  ( (  Hom  `  ( A  X.c  C ) ) `  x ) )  -> 
g  e.  ( ( ( 1st `  ( 2nd `  x ) ) (  Hom  `  A
) ( 1st `  y
) )  X.  (
( 2nd `  ( 2nd `  x ) ) (  Hom  `  C
) ( 2nd `  y
) ) ) )
75 xp1st 6191 . . . . . . . . . 10  |-  ( g  e.  ( ( ( 1st `  ( 2nd `  x ) ) (  Hom  `  A )
( 1st `  y
) )  X.  (
( 2nd `  ( 2nd `  x ) ) (  Hom  `  C
) ( 2nd `  y
) ) )  -> 
( 1st `  g
)  e.  ( ( 1st `  ( 2nd `  x ) ) (  Hom  `  A )
( 1st `  y
) ) )
7674, 75syl 15 . . . . . . . . 9  |-  ( ( ( ( ( ph  /\  x  e.  ( ( ( Base `  A
)  X.  ( Base `  C ) )  X.  ( ( Base `  A
)  X.  ( Base `  C ) ) ) )  /\  y  e.  ( ( Base `  A
)  X.  ( Base `  C ) ) )  /\  g  e.  ( ( 2nd `  x
) (  Hom  `  ( A  X.c  C ) ) y ) )  /\  f  e.  ( (  Hom  `  ( A  X.c  C ) ) `  x ) )  -> 
( 1st `  g
)  e.  ( ( 1st `  ( 2nd `  x ) ) (  Hom  `  A )
( 1st `  y
) ) )
772, 4, 6, 22, 46, 48, 53, 57, 60, 71, 76comfeqval 13660 . . . . . . . 8  |-  ( ( ( ( ( ph  /\  x  e.  ( ( ( Base `  A
)  X.  ( Base `  C ) )  X.  ( ( Base `  A
)  X.  ( Base `  C ) ) ) )  /\  y  e.  ( ( Base `  A
)  X.  ( Base `  C ) ) )  /\  g  e.  ( ( 2nd `  x
) (  Hom  `  ( A  X.c  C ) ) y ) )  /\  f  e.  ( (  Hom  `  ( A  X.c  C ) ) `  x ) )  -> 
( ( 1st `  g
) ( <. ( 1st `  ( 1st `  x
) ) ,  ( 1st `  ( 2nd `  x ) ) >.
(comp `  A )
( 1st `  y
) ) ( 1st `  f ) )  =  ( ( 1st `  g
) ( <. ( 1st `  ( 1st `  x
) ) ,  ( 1st `  ( 2nd `  x ) ) >.
(comp `  B )
( 1st `  y
) ) ( 1st `  f ) ) )
7828ad4antr 712 . . . . . . . . 9  |-  ( ( ( ( ( ph  /\  x  e.  ( ( ( Base `  A
)  X.  ( Base `  C ) )  X.  ( ( Base `  A
)  X.  ( Base `  C ) ) ) )  /\  y  e.  ( ( Base `  A
)  X.  ( Base `  C ) ) )  /\  g  e.  ( ( 2nd `  x
) (  Hom  `  ( A  X.c  C ) ) y ) )  /\  f  e.  ( (  Hom  `  ( A  X.c  C ) ) `  x ) )  -> 
(  Homf 
`  C )  =  (  Homf 
`  D ) )
79 xpcpropd.4 . . . . . . . . . 10  |-  ( ph  ->  (compf `  C )  =  (compf `  D ) )
8079ad4antr 712 . . . . . . . . 9  |-  ( ( ( ( ( ph  /\  x  e.  ( ( ( Base `  A
)  X.  ( Base `  C ) )  X.  ( ( Base `  A
)  X.  ( Base `  C ) ) ) )  /\  y  e.  ( ( Base `  A
)  X.  ( Base `  C ) ) )  /\  g  e.  ( ( 2nd `  x
) (  Hom  `  ( A  X.c  C ) ) y ) )  /\  f  e.  ( (  Hom  `  ( A  X.c  C ) ) `  x ) )  -> 
(compf `  C )  =  (compf `  D ) )
81 xp2nd 6192 . . . . . . . . . 10  |-  ( ( 1st `  x )  e.  ( ( Base `  A )  X.  ( Base `  C ) )  ->  ( 2nd `  ( 1st `  x ) )  e.  ( Base `  C
) )
8251, 81syl 15 . . . . . . . . 9  |-  ( ( ( ( ( ph  /\  x  e.  ( ( ( Base `  A
)  X.  ( Base `  C ) )  X.  ( ( Base `  A
)  X.  ( Base `  C ) ) ) )  /\  y  e.  ( ( Base `  A
)  X.  ( Base `  C ) ) )  /\  g  e.  ( ( 2nd `  x
) (  Hom  `  ( A  X.c  C ) ) y ) )  /\  f  e.  ( (  Hom  `  ( A  X.c  C ) ) `  x ) )  -> 
( 2nd `  ( 1st `  x ) )  e.  ( Base `  C
) )
83 xp2nd 6192 . . . . . . . . . 10  |-  ( ( 2nd `  x )  e.  ( ( Base `  A )  X.  ( Base `  C ) )  ->  ( 2nd `  ( 2nd `  x ) )  e.  ( Base `  C
) )
8455, 83syl 15 . . . . . . . . 9  |-  ( ( ( ( ( ph  /\  x  e.  ( ( ( Base `  A
)  X.  ( Base `  C ) )  X.  ( ( Base `  A
)  X.  ( Base `  C ) ) ) )  /\  y  e.  ( ( Base `  A
)  X.  ( Base `  C ) ) )  /\  g  e.  ( ( 2nd `  x
) (  Hom  `  ( A  X.c  C ) ) y ) )  /\  f  e.  ( (  Hom  `  ( A  X.c  C ) ) `  x ) )  -> 
( 2nd `  ( 2nd `  x ) )  e.  ( Base `  C
) )
85 xp2nd 6192 . . . . . . . . . 10  |-  ( y  e.  ( ( Base `  A )  X.  ( Base `  C ) )  ->  ( 2nd `  y
)  e.  ( Base `  C ) )
8658, 85syl 15 . . . . . . . . 9  |-  ( ( ( ( ( ph  /\  x  e.  ( ( ( Base `  A
)  X.  ( Base `  C ) )  X.  ( ( Base `  A
)  X.  ( Base `  C ) ) ) )  /\  y  e.  ( ( Base `  A
)  X.  ( Base `  C ) ) )  /\  g  e.  ( ( 2nd `  x
) (  Hom  `  ( A  X.c  C ) ) y ) )  /\  f  e.  ( (  Hom  `  ( A  X.c  C ) ) `  x ) )  -> 
( 2nd `  y
)  e.  ( Base `  C ) )
87 xp2nd 6192 . . . . . . . . . 10  |-  ( f  e.  ( ( ( 1st `  ( 1st `  x ) ) (  Hom  `  A )
( 1st `  ( 2nd `  x ) ) )  X.  ( ( 2nd `  ( 1st `  x ) ) (  Hom  `  C )
( 2nd `  ( 2nd `  x ) ) ) )  ->  ( 2nd `  f )  e.  ( ( 2nd `  ( 1st `  x ) ) (  Hom  `  C
) ( 2nd `  ( 2nd `  x ) ) ) )
8869, 87syl 15 . . . . . . . . 9  |-  ( ( ( ( ( ph  /\  x  e.  ( ( ( Base `  A
)  X.  ( Base `  C ) )  X.  ( ( Base `  A
)  X.  ( Base `  C ) ) ) )  /\  y  e.  ( ( Base `  A
)  X.  ( Base `  C ) ) )  /\  g  e.  ( ( 2nd `  x
) (  Hom  `  ( A  X.c  C ) ) y ) )  /\  f  e.  ( (  Hom  `  ( A  X.c  C ) ) `  x ) )  -> 
( 2nd `  f
)  e.  ( ( 2nd `  ( 1st `  x ) ) (  Hom  `  C )
( 2nd `  ( 2nd `  x ) ) ) )
89 xp2nd 6192 . . . . . . . . . 10  |-  ( g  e.  ( ( ( 1st `  ( 2nd `  x ) ) (  Hom  `  A )
( 1st `  y
) )  X.  (
( 2nd `  ( 2nd `  x ) ) (  Hom  `  C
) ( 2nd `  y
) ) )  -> 
( 2nd `  g
)  e.  ( ( 2nd `  ( 2nd `  x ) ) (  Hom  `  C )
( 2nd `  y
) ) )
9074, 89syl 15 . . . . . . . . 9  |-  ( ( ( ( ( ph  /\  x  e.  ( ( ( Base `  A
)  X.  ( Base `  C ) )  X.  ( ( Base `  A
)  X.  ( Base `  C ) ) ) )  /\  y  e.  ( ( Base `  A
)  X.  ( Base `  C ) ) )  /\  g  e.  ( ( 2nd `  x
) (  Hom  `  ( A  X.c  C ) ) y ) )  /\  f  e.  ( (  Hom  `  ( A  X.c  C ) ) `  x ) )  -> 
( 2nd `  g
)  e.  ( ( 2nd `  ( 2nd `  x ) ) (  Hom  `  C )
( 2nd `  y
) ) )
913, 5, 7, 23, 78, 80, 82, 84, 86, 88, 90comfeqval 13660 . . . . . . . 8  |-  ( ( ( ( ( ph  /\  x  e.  ( ( ( Base `  A
)  X.  ( Base `  C ) )  X.  ( ( Base `  A
)  X.  ( Base `  C ) ) ) )  /\  y  e.  ( ( Base `  A
)  X.  ( Base `  C ) ) )  /\  g  e.  ( ( 2nd `  x
) (  Hom  `  ( A  X.c  C ) ) y ) )  /\  f  e.  ( (  Hom  `  ( A  X.c  C ) ) `  x ) )  -> 
( ( 2nd `  g
) ( <. ( 2nd `  ( 1st `  x
) ) ,  ( 2nd `  ( 2nd `  x ) ) >.
(comp `  C )
( 2nd `  y
) ) ( 2nd `  f ) )  =  ( ( 2nd `  g
) ( <. ( 2nd `  ( 1st `  x
) ) ,  ( 2nd `  ( 2nd `  x ) ) >.
(comp `  D )
( 2nd `  y
) ) ( 2nd `  f ) ) )
9277, 91opeq12d 3841 . . . . . . 7  |-  ( ( ( ( ( ph  /\  x  e.  ( ( ( Base `  A
)  X.  ( Base `  C ) )  X.  ( ( Base `  A
)  X.  ( Base `  C ) ) ) )  /\  y  e.  ( ( Base `  A
)  X.  ( Base `  C ) ) )  /\  g  e.  ( ( 2nd `  x
) (  Hom  `  ( A  X.c  C ) ) y ) )  /\  f  e.  ( (  Hom  `  ( A  X.c  C ) ) `  x ) )  ->  <. ( ( 1st `  g
) ( <. ( 1st `  ( 1st `  x
) ) ,  ( 1st `  ( 2nd `  x ) ) >.
(comp `  A )
( 1st `  y
) ) ( 1st `  f ) ) ,  ( ( 2nd `  g
) ( <. ( 2nd `  ( 1st `  x
) ) ,  ( 2nd `  ( 2nd `  x ) ) >.
(comp `  C )
( 2nd `  y
) ) ( 2nd `  f ) ) >.  =  <. ( ( 1st `  g ) ( <.
( 1st `  ( 1st `  x ) ) ,  ( 1st `  ( 2nd `  x ) )
>. (comp `  B )
( 1st `  y
) ) ( 1st `  f ) ) ,  ( ( 2nd `  g
) ( <. ( 2nd `  ( 1st `  x
) ) ,  ( 2nd `  ( 2nd `  x ) ) >.
(comp `  D )
( 2nd `  y
) ) ( 2nd `  f ) ) >.
)
93923impa 1146 . . . . . 6  |-  ( ( ( ( ph  /\  x  e.  ( (
( Base `  A )  X.  ( Base `  C
) )  X.  (
( Base `  A )  X.  ( Base `  C
) ) ) )  /\  y  e.  ( ( Base `  A
)  X.  ( Base `  C ) ) )  /\  g  e.  ( ( 2nd `  x
) (  Hom  `  ( A  X.c  C ) ) y )  /\  f  e.  ( (  Hom  `  ( A  X.c  C ) ) `  x ) )  ->  <. ( ( 1st `  g
) ( <. ( 1st `  ( 1st `  x
) ) ,  ( 1st `  ( 2nd `  x ) ) >.
(comp `  A )
( 1st `  y
) ) ( 1st `  f ) ) ,  ( ( 2nd `  g
) ( <. ( 2nd `  ( 1st `  x
) ) ,  ( 2nd `  ( 2nd `  x ) ) >.
(comp `  C )
( 2nd `  y
) ) ( 2nd `  f ) ) >.  =  <. ( ( 1st `  g ) ( <.
( 1st `  ( 1st `  x ) ) ,  ( 1st `  ( 2nd `  x ) )
>. (comp `  B )
( 1st `  y
) ) ( 1st `  f ) ) ,  ( ( 2nd `  g
) ( <. ( 2nd `  ( 1st `  x
) ) ,  ( 2nd `  ( 2nd `  x ) ) >.
(comp `  D )
( 2nd `  y
) ) ( 2nd `  f ) ) >.
)
9493mpt2eq3dva 5954 . . . . 5  |-  ( ( ( ph  /\  x  e.  ( ( ( Base `  A )  X.  ( Base `  C ) )  X.  ( ( Base `  A )  X.  ( Base `  C ) ) ) )  /\  y  e.  ( ( Base `  A
)  X.  ( Base `  C ) ) )  ->  ( g  e.  ( ( 2nd `  x
) (  Hom  `  ( A  X.c  C ) ) y ) ,  f  e.  ( (  Hom  `  ( A  X.c  C ) ) `  x )  |->  <. (
( 1st `  g
) ( <. ( 1st `  ( 1st `  x
) ) ,  ( 1st `  ( 2nd `  x ) ) >.
(comp `  A )
( 1st `  y
) ) ( 1st `  f ) ) ,  ( ( 2nd `  g
) ( <. ( 2nd `  ( 1st `  x
) ) ,  ( 2nd `  ( 2nd `  x ) ) >.
(comp `  C )
( 2nd `  y
) ) ( 2nd `  f ) ) >.
)  =  ( g  e.  ( ( 2nd `  x ) (  Hom  `  ( A  X.c  C ) ) y ) ,  f  e.  ( (  Hom  `  ( A  X.c  C ) ) `  x )  |->  <. (
( 1st `  g
) ( <. ( 1st `  ( 1st `  x
) ) ,  ( 1st `  ( 2nd `  x ) ) >.
(comp `  B )
( 1st `  y
) ) ( 1st `  f ) ) ,  ( ( 2nd `  g
) ( <. ( 2nd `  ( 1st `  x
) ) ,  ( 2nd `  ( 2nd `  x ) ) >.
(comp `  D )
( 2nd `  y
) ) ( 2nd `  f ) ) >.
) )
95943impa 1146 . . . 4  |-  ( (
ph  /\  x  e.  ( ( ( Base `  A )  X.  ( Base `  C ) )  X.  ( ( Base `  A )  X.  ( Base `  C ) ) )  /\  y  e.  ( ( Base `  A
)  X.  ( Base `  C ) ) )  ->  ( g  e.  ( ( 2nd `  x
) (  Hom  `  ( A  X.c  C ) ) y ) ,  f  e.  ( (  Hom  `  ( A  X.c  C ) ) `  x )  |->  <. (
( 1st `  g
) ( <. ( 1st `  ( 1st `  x
) ) ,  ( 1st `  ( 2nd `  x ) ) >.
(comp `  A )
( 1st `  y
) ) ( 1st `  f ) ) ,  ( ( 2nd `  g
) ( <. ( 2nd `  ( 1st `  x
) ) ,  ( 2nd `  ( 2nd `  x ) ) >.
(comp `  C )
( 2nd `  y
) ) ( 2nd `  f ) ) >.
)  =  ( g  e.  ( ( 2nd `  x ) (  Hom  `  ( A  X.c  C ) ) y ) ,  f  e.  ( (  Hom  `  ( A  X.c  C ) ) `  x )  |->  <. (
( 1st `  g
) ( <. ( 1st `  ( 1st `  x
) ) ,  ( 1st `  ( 2nd `  x ) ) >.
(comp `  B )
( 1st `  y
) ) ( 1st `  f ) ) ,  ( ( 2nd `  g
) ( <. ( 2nd `  ( 1st `  x
) ) ,  ( 2nd `  ( 2nd `  x ) ) >.
(comp `  D )
( 2nd `  y
) ) ( 2nd `  f ) ) >.
) )
9695mpt2eq3dva 5954 . . 3  |-  ( ph  ->  ( x  e.  ( ( ( Base `  A
)  X.  ( Base `  C ) )  X.  ( ( Base `  A
)  X.  ( Base `  C ) ) ) ,  y  e.  ( ( Base `  A
)  X.  ( Base `  C ) )  |->  ( g  e.  ( ( 2nd `  x ) (  Hom  `  ( A  X.c  C ) ) y ) ,  f  e.  ( (  Hom  `  ( A  X.c  C ) ) `  x )  |->  <. (
( 1st `  g
) ( <. ( 1st `  ( 1st `  x
) ) ,  ( 1st `  ( 2nd `  x ) ) >.
(comp `  A )
( 1st `  y
) ) ( 1st `  f ) ) ,  ( ( 2nd `  g
) ( <. ( 2nd `  ( 1st `  x
) ) ,  ( 2nd `  ( 2nd `  x ) ) >.
(comp `  C )
( 2nd `  y
) ) ( 2nd `  f ) ) >.
) )  =  ( x  e.  ( ( ( Base `  A
)  X.  ( Base `  C ) )  X.  ( ( Base `  A
)  X.  ( Base `  C ) ) ) ,  y  e.  ( ( Base `  A
)  X.  ( Base `  C ) )  |->  ( g  e.  ( ( 2nd `  x ) (  Hom  `  ( A  X.c  C ) ) y ) ,  f  e.  ( (  Hom  `  ( A  X.c  C ) ) `  x )  |->  <. (
( 1st `  g
) ( <. ( 1st `  ( 1st `  x
) ) ,  ( 1st `  ( 2nd `  x ) ) >.
(comp `  B )
( 1st `  y
) ) ( 1st `  f ) ) ,  ( ( 2nd `  g
) ( <. ( 2nd `  ( 1st `  x
) ) ,  ( 2nd `  ( 2nd `  x ) ) >.
(comp `  D )
( 2nd `  y
) ) ( 2nd `  f ) ) >.
) ) )
9717, 18, 19, 20, 21, 22, 23, 24, 25, 30, 45, 96xpcval 14000 . 2  |-  ( ph  ->  ( B  X.c  D )  =  { <. ( Base `  ndx ) ,  ( ( Base `  A
)  X.  ( Base `  C ) ) >. ,  <. (  Hom  `  ndx ) ,  (  Hom  `  ( A  X.c  C ) ) >. ,  <. (comp ` 
ndx ) ,  ( x  e.  ( ( ( Base `  A
)  X.  ( Base `  C ) )  X.  ( ( Base `  A
)  X.  ( Base `  C ) ) ) ,  y  e.  ( ( Base `  A
)  X.  ( Base `  C ) )  |->  ( g  e.  ( ( 2nd `  x ) (  Hom  `  ( A  X.c  C ) ) y ) ,  f  e.  ( (  Hom  `  ( A  X.c  C ) ) `  x )  |->  <. (
( 1st `  g
) ( <. ( 1st `  ( 1st `  x
) ) ,  ( 1st `  ( 2nd `  x ) ) >.
(comp `  A )
( 1st `  y
) ) ( 1st `  f ) ) ,  ( ( 2nd `  g
) ( <. ( 2nd `  ( 1st `  x
) ) ,  ( 2nd `  ( 2nd `  x ) ) >.
(comp `  C )
( 2nd `  y
) ) ( 2nd `  f ) ) >.
) ) >. } )
9816, 97eqtr4d 2351 1  |-  ( ph  ->  ( A  X.c  C )  =  ( B  X.c  D
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1633    e. wcel 1701   {ctp 3676   <.cop 3677    X. cxp 4724   ` cfv 5292  (class class class)co 5900    e. cmpt2 5902   1stc1st 6162   2ndc2nd 6163   ndxcnx 13192   Basecbs 13195    Hom chom 13266  compcco 13267    Homf chomf 13617  compfccomf 13618    X.c cxpc 13991
This theorem is referenced by:  curfpropd  14056  oppchofcl  14083
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549  ax-cnex 8838  ax-resscn 8839  ax-1cn 8840  ax-icn 8841  ax-addcl 8842  ax-addrcl 8843  ax-mulcl 8844  ax-mulrcl 8845  ax-mulcom 8846  ax-addass 8847  ax-mulass 8848  ax-distr 8849  ax-i2m1 8850  ax-1ne0 8851  ax-1rid 8852  ax-rnegex 8853  ax-rrecex 8854  ax-cnre 8855  ax-pre-lttri 8856  ax-pre-lttrn 8857  ax-pre-ltadd 8858  ax-pre-mulgt0 8859
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-nel 2482  df-ral 2582  df-rex 2583  df-reu 2584  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-pss 3202  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-tp 3682  df-op 3683  df-uni 3865  df-int 3900  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-tr 4151  df-eprel 4342  df-id 4346  df-po 4351  df-so 4352  df-fr 4389  df-we 4391  df-ord 4432  df-on 4433  df-lim 4434  df-suc 4435  df-om 4694  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-1st 6164  df-2nd 6165  df-riota 6346  df-recs 6430  df-rdg 6465  df-1o 6521  df-oadd 6525  df-er 6702  df-en 6907  df-dom 6908  df-sdom 6909  df-fin 6910  df-pnf 8914  df-mnf 8915  df-xr 8916  df-ltxr 8917  df-le 8918  df-sub 9084  df-neg 9085  df-nn 9792  df-2 9849  df-3 9850  df-4 9851  df-5 9852  df-6 9853  df-7 9854  df-8 9855  df-9 9856  df-10 9857  df-n0 10013  df-z 10072  df-dec 10172  df-uz 10278  df-fz 10830  df-struct 13197  df-ndx 13198  df-slot 13199  df-base 13200  df-hom 13279  df-cco 13280  df-homf 13621  df-comf 13622  df-xpc 13995
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