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Theorem comfeq 14637
Description: Condition for two categories with the same hom-sets to have the same composition. (Contributed by Mario Carneiro, 4-Jan-2017.)
Hypotheses
Ref Expression
comfeq.1  |-  .x.  =  (comp `  C )
comfeq.2  |-  .xb  =  (comp `  D )
comfeq.h  |-  H  =  ( Hom  `  C
)
comfeq.3  |-  ( ph  ->  B  =  ( Base `  C ) )
comfeq.4  |-  ( ph  ->  B  =  ( Base `  D ) )
comfeq.5  |-  ( ph  ->  ( Hom f  `  C )  =  ( Hom f  `  D ) )
Assertion
Ref Expression
comfeq  |-  ( ph  ->  ( (compf `  C )  =  (compf `  D )  <->  A. x  e.  B  A. y  e.  B  A. z  e.  B  A. f  e.  ( x H y ) A. g  e.  ( y H z ) ( g (
<. x ,  y >.  .x.  z ) f )  =  ( g (
<. x ,  y >.  .xb  z ) f ) ) )
Distinct variable groups:    f, g, x, y, z, B    C, f, g, z    ph, f,
g, z    .x. , f, g, x, y    D, f, g, z    f, H, g, x, y    .xb , f,
g, x, y
Allowed substitution hints:    ph( x, y)    C( x, y)    D( x, y)    .xb ( z)    .x. ( z)    H( z)

Proof of Theorem comfeq
Dummy variable  u is distinct from all other variables.
StepHypRef Expression
1 ovex 6111 . . . . . 6  |-  ( ( 2nd `  u ) H z )  e. 
_V
2 fvex 5696 . . . . . 6  |-  ( H `
 u )  e. 
_V
31, 2mpt2ex 6645 . . . . 5  |-  ( g  e.  ( ( 2nd `  u ) H z ) ,  f  e.  ( H `  u
)  |->  ( g ( u  .x.  z ) f ) )  e. 
_V
43rgen2w 2779 . . . 4  |-  A. u  e.  ( B  X.  B
) A. z  e.  B  ( g  e.  ( ( 2nd `  u
) H z ) ,  f  e.  ( H `  u ) 
|->  ( g ( u 
.x.  z ) f ) )  e.  _V
5 mpt22eqb 6194 . . . 4  |-  ( A. u  e.  ( B  X.  B ) A. z  e.  B  ( g  e.  ( ( 2nd `  u
) H z ) ,  f  e.  ( H `  u ) 
|->  ( g ( u 
.x.  z ) f ) )  e.  _V  ->  ( ( u  e.  ( B  X.  B
) ,  z  e.  B  |->  ( g  e.  ( ( 2nd `  u
) H z ) ,  f  e.  ( H `  u ) 
|->  ( g ( u 
.x.  z ) f ) ) )  =  ( u  e.  ( B  X.  B ) ,  z  e.  B  |->  ( g  e.  ( ( 2nd `  u
) H z ) ,  f  e.  ( H `  u ) 
|->  ( g ( u 
.xb  z ) f ) ) )  <->  A. u  e.  ( B  X.  B
) A. z  e.  B  ( g  e.  ( ( 2nd `  u
) H z ) ,  f  e.  ( H `  u ) 
|->  ( g ( u 
.x.  z ) f ) )  =  ( g  e.  ( ( 2nd `  u ) H z ) ,  f  e.  ( H `
 u )  |->  ( g ( u  .xb  z ) f ) ) ) )
64, 5ax-mp 5 . . 3  |-  ( ( u  e.  ( B  X.  B ) ,  z  e.  B  |->  ( g  e.  ( ( 2nd `  u ) H z ) ,  f  e.  ( H `
 u )  |->  ( g ( u  .x.  z ) f ) ) )  =  ( u  e.  ( B  X.  B ) ,  z  e.  B  |->  ( g  e.  ( ( 2nd `  u ) H z ) ,  f  e.  ( H `
 u )  |->  ( g ( u  .xb  z ) f ) ) )  <->  A. u  e.  ( B  X.  B
) A. z  e.  B  ( g  e.  ( ( 2nd `  u
) H z ) ,  f  e.  ( H `  u ) 
|->  ( g ( u 
.x.  z ) f ) )  =  ( g  e.  ( ( 2nd `  u ) H z ) ,  f  e.  ( H `
 u )  |->  ( g ( u  .xb  z ) f ) ) )
7 vex 2970 . . . . . . . . 9  |-  x  e. 
_V
8 vex 2970 . . . . . . . . 9  |-  y  e. 
_V
97, 8op2ndd 6583 . . . . . . . 8  |-  ( u  =  <. x ,  y
>.  ->  ( 2nd `  u
)  =  y )
109oveq1d 6101 . . . . . . 7  |-  ( u  =  <. x ,  y
>.  ->  ( ( 2nd `  u ) H z )  =  ( y H z ) )
11 fveq2 5686 . . . . . . . . 9  |-  ( u  =  <. x ,  y
>.  ->  ( H `  u )  =  ( H `  <. x ,  y >. )
)
12 df-ov 6089 . . . . . . . . 9  |-  ( x H y )  =  ( H `  <. x ,  y >. )
1311, 12syl6eqr 2488 . . . . . . . 8  |-  ( u  =  <. x ,  y
>.  ->  ( H `  u )  =  ( x H y ) )
14 oveq1 6093 . . . . . . . . . 10  |-  ( u  =  <. x ,  y
>.  ->  ( u  .x.  z )  =  (
<. x ,  y >.  .x.  z ) )
1514oveqd 6103 . . . . . . . . 9  |-  ( u  =  <. x ,  y
>.  ->  ( g ( u  .x.  z ) f )  =  ( g ( <. x ,  y >.  .x.  z
) f ) )
16 oveq1 6093 . . . . . . . . . 10  |-  ( u  =  <. x ,  y
>.  ->  ( u  .xb  z )  =  (
<. x ,  y >.  .xb  z ) )
1716oveqd 6103 . . . . . . . . 9  |-  ( u  =  <. x ,  y
>.  ->  ( g ( u  .xb  z )
f )  =  ( g ( <. x ,  y >.  .xb  z
) f ) )
1815, 17eqeq12d 2452 . . . . . . . 8  |-  ( u  =  <. x ,  y
>.  ->  ( ( g ( u  .x.  z
) f )  =  ( g ( u 
.xb  z ) f )  <->  ( g (
<. x ,  y >.  .x.  z ) f )  =  ( g (
<. x ,  y >.  .xb  z ) f ) ) )
1913, 18raleqbidv 2926 . . . . . . 7  |-  ( u  =  <. x ,  y
>.  ->  ( A. f  e.  ( H `  u
) ( g ( u  .x.  z ) f )  =  ( g ( u  .xb  z ) f )  <->  A. f  e.  (
x H y ) ( g ( <.
x ,  y >.  .x.  z ) f )  =  ( g (
<. x ,  y >.  .xb  z ) f ) ) )
2010, 19raleqbidv 2926 . . . . . 6  |-  ( u  =  <. x ,  y
>.  ->  ( A. g  e.  ( ( 2nd `  u
) H z ) A. f  e.  ( H `  u ) ( g ( u 
.x.  z ) f )  =  ( g ( u  .xb  z
) f )  <->  A. g  e.  ( y H z ) A. f  e.  ( x H y ) ( g (
<. x ,  y >.  .x.  z ) f )  =  ( g (
<. x ,  y >.  .xb  z ) f ) ) )
21 ovex 6111 . . . . . . . 8  |-  ( g ( u  .x.  z
) f )  e. 
_V
2221rgen2w 2779 . . . . . . 7  |-  A. g  e.  ( ( 2nd `  u
) H z ) A. f  e.  ( H `  u ) ( g ( u 
.x.  z ) f )  e.  _V
23 mpt22eqb 6194 . . . . . . 7  |-  ( A. g  e.  ( ( 2nd `  u ) H z ) A. f  e.  ( H `  u
) ( g ( u  .x.  z ) f )  e.  _V  ->  ( ( g  e.  ( ( 2nd `  u
) H z ) ,  f  e.  ( H `  u ) 
|->  ( g ( u 
.x.  z ) f ) )  =  ( g  e.  ( ( 2nd `  u ) H z ) ,  f  e.  ( H `
 u )  |->  ( g ( u  .xb  z ) f ) )  <->  A. g  e.  ( ( 2nd `  u
) H z ) A. f  e.  ( H `  u ) ( g ( u 
.x.  z ) f )  =  ( g ( u  .xb  z
) f ) ) )
2422, 23ax-mp 5 . . . . . 6  |-  ( ( g  e.  ( ( 2nd `  u ) H z ) ,  f  e.  ( H `
 u )  |->  ( g ( u  .x.  z ) f ) )  =  ( g  e.  ( ( 2nd `  u ) H z ) ,  f  e.  ( H `  u
)  |->  ( g ( u  .xb  z )
f ) )  <->  A. g  e.  ( ( 2nd `  u
) H z ) A. f  e.  ( H `  u ) ( g ( u 
.x.  z ) f )  =  ( g ( u  .xb  z
) f ) )
25 ralcom 2876 . . . . . 6  |-  ( A. f  e.  ( x H y ) A. g  e.  ( y H z ) ( g ( <. x ,  y >.  .x.  z
) f )  =  ( g ( <.
x ,  y >.  .xb  z ) f )  <->  A. g  e.  (
y H z ) A. f  e.  ( x H y ) ( g ( <.
x ,  y >.  .x.  z ) f )  =  ( g (
<. x ,  y >.  .xb  z ) f ) )
2620, 24, 253bitr4g 288 . . . . 5  |-  ( u  =  <. x ,  y
>.  ->  ( ( g  e.  ( ( 2nd `  u ) H z ) ,  f  e.  ( H `  u
)  |->  ( g ( u  .x.  z ) f ) )  =  ( g  e.  ( ( 2nd `  u
) H z ) ,  f  e.  ( H `  u ) 
|->  ( g ( u 
.xb  z ) f ) )  <->  A. f  e.  ( x H y ) A. g  e.  ( y H z ) ( g (
<. x ,  y >.  .x.  z ) f )  =  ( g (
<. x ,  y >.  .xb  z ) f ) ) )
2726ralbidv 2730 . . . 4  |-  ( u  =  <. x ,  y
>.  ->  ( A. z  e.  B  ( g  e.  ( ( 2nd `  u
) H z ) ,  f  e.  ( H `  u ) 
|->  ( g ( u 
.x.  z ) f ) )  =  ( g  e.  ( ( 2nd `  u ) H z ) ,  f  e.  ( H `
 u )  |->  ( g ( u  .xb  z ) f ) )  <->  A. z  e.  B  A. f  e.  (
x H y ) A. g  e.  ( y H z ) ( g ( <.
x ,  y >.  .x.  z ) f )  =  ( g (
<. x ,  y >.  .xb  z ) f ) ) )
2827ralxp 4976 . . 3  |-  ( A. u  e.  ( B  X.  B ) A. z  e.  B  ( g  e.  ( ( 2nd `  u
) H z ) ,  f  e.  ( H `  u ) 
|->  ( g ( u 
.x.  z ) f ) )  =  ( g  e.  ( ( 2nd `  u ) H z ) ,  f  e.  ( H `
 u )  |->  ( g ( u  .xb  z ) f ) )  <->  A. x  e.  B  A. y  e.  B  A. z  e.  B  A. f  e.  (
x H y ) A. g  e.  ( y H z ) ( g ( <.
x ,  y >.  .x.  z ) f )  =  ( g (
<. x ,  y >.  .xb  z ) f ) )
296, 28bitri 249 . 2  |-  ( ( u  e.  ( B  X.  B ) ,  z  e.  B  |->  ( g  e.  ( ( 2nd `  u ) H z ) ,  f  e.  ( H `
 u )  |->  ( g ( u  .x.  z ) f ) ) )  =  ( u  e.  ( B  X.  B ) ,  z  e.  B  |->  ( g  e.  ( ( 2nd `  u ) H z ) ,  f  e.  ( H `
 u )  |->  ( g ( u  .xb  z ) f ) ) )  <->  A. x  e.  B  A. y  e.  B  A. z  e.  B  A. f  e.  ( x H y ) A. g  e.  ( y H z ) ( g (
<. x ,  y >.  .x.  z ) f )  =  ( g (
<. x ,  y >.  .xb  z ) f ) )
30 comfeq.3 . . . . . 6  |-  ( ph  ->  B  =  ( Base `  C ) )
3130, 30xpeq12d 4860 . . . . 5  |-  ( ph  ->  ( B  X.  B
)  =  ( (
Base `  C )  X.  ( Base `  C
) ) )
32 eqidd 2439 . . . . 5  |-  ( ph  ->  ( g  e.  ( ( 2nd `  u
) H z ) ,  f  e.  ( H `  u ) 
|->  ( g ( u 
.x.  z ) f ) )  =  ( g  e.  ( ( 2nd `  u ) H z ) ,  f  e.  ( H `
 u )  |->  ( g ( u  .x.  z ) f ) ) )
3331, 30, 32mpt2eq123dv 6143 . . . 4  |-  ( ph  ->  ( u  e.  ( B  X.  B ) ,  z  e.  B  |->  ( g  e.  ( ( 2nd `  u
) H z ) ,  f  e.  ( H `  u ) 
|->  ( g ( u 
.x.  z ) f ) ) )  =  ( u  e.  ( ( Base `  C
)  X.  ( Base `  C ) ) ,  z  e.  ( Base `  C )  |->  ( g  e.  ( ( 2nd `  u ) H z ) ,  f  e.  ( H `  u
)  |->  ( g ( u  .x.  z ) f ) ) ) )
34 eqid 2438 . . . . 5  |-  (compf `  C
)  =  (compf `  C
)
35 eqid 2438 . . . . 5  |-  ( Base `  C )  =  (
Base `  C )
36 comfeq.h . . . . 5  |-  H  =  ( Hom  `  C
)
37 comfeq.1 . . . . 5  |-  .x.  =  (comp `  C )
3834, 35, 36, 37comfffval 14629 . . . 4  |-  (compf `  C
)  =  ( u  e.  ( ( Base `  C )  X.  ( Base `  C ) ) ,  z  e.  (
Base `  C )  |->  ( g  e.  ( ( 2nd `  u
) H z ) ,  f  e.  ( H `  u ) 
|->  ( g ( u 
.x.  z ) f ) ) )
3933, 38syl6eqr 2488 . . 3  |-  ( ph  ->  ( u  e.  ( B  X.  B ) ,  z  e.  B  |->  ( g  e.  ( ( 2nd `  u
) H z ) ,  f  e.  ( H `  u ) 
|->  ( g ( u 
.x.  z ) f ) ) )  =  (compf `  C ) )
40 eqid 2438 . . . . . . . 8  |-  ( Hom  `  D )  =  ( Hom  `  D )
41 comfeq.5 . . . . . . . . 9  |-  ( ph  ->  ( Hom f  `  C )  =  ( Hom f  `  D ) )
42413ad2ant1 1009 . . . . . . . 8  |-  ( (
ph  /\  u  e.  ( B  X.  B
)  /\  z  e.  B )  ->  ( Hom f  `  C )  =  ( Hom f  `  D ) )
43 xp2nd 6602 . . . . . . . . . 10  |-  ( u  e.  ( B  X.  B )  ->  ( 2nd `  u )  e.  B )
44433ad2ant2 1010 . . . . . . . . 9  |-  ( (
ph  /\  u  e.  ( B  X.  B
)  /\  z  e.  B )  ->  ( 2nd `  u )  e.  B )
45303ad2ant1 1009 . . . . . . . . 9  |-  ( (
ph  /\  u  e.  ( B  X.  B
)  /\  z  e.  B )  ->  B  =  ( Base `  C
) )
4644, 45eleqtrd 2514 . . . . . . . 8  |-  ( (
ph  /\  u  e.  ( B  X.  B
)  /\  z  e.  B )  ->  ( 2nd `  u )  e.  ( Base `  C
) )
47 simp3 990 . . . . . . . . 9  |-  ( (
ph  /\  u  e.  ( B  X.  B
)  /\  z  e.  B )  ->  z  e.  B )
4847, 45eleqtrd 2514 . . . . . . . 8  |-  ( (
ph  /\  u  e.  ( B  X.  B
)  /\  z  e.  B )  ->  z  e.  ( Base `  C
) )
4935, 36, 40, 42, 46, 48homfeqval 14628 . . . . . . 7  |-  ( (
ph  /\  u  e.  ( B  X.  B
)  /\  z  e.  B )  ->  (
( 2nd `  u
) H z )  =  ( ( 2nd `  u ) ( Hom  `  D ) z ) )
50 xp1st 6601 . . . . . . . . . . . 12  |-  ( u  e.  ( B  X.  B )  ->  ( 1st `  u )  e.  B )
51503ad2ant2 1010 . . . . . . . . . . 11  |-  ( (
ph  /\  u  e.  ( B  X.  B
)  /\  z  e.  B )  ->  ( 1st `  u )  e.  B )
5251, 45eleqtrd 2514 . . . . . . . . . 10  |-  ( (
ph  /\  u  e.  ( B  X.  B
)  /\  z  e.  B )  ->  ( 1st `  u )  e.  ( Base `  C
) )
5335, 36, 40, 42, 52, 46homfeqval 14628 . . . . . . . . 9  |-  ( (
ph  /\  u  e.  ( B  X.  B
)  /\  z  e.  B )  ->  (
( 1st `  u
) H ( 2nd `  u ) )  =  ( ( 1st `  u
) ( Hom  `  D
) ( 2nd `  u
) ) )
54 df-ov 6089 . . . . . . . . 9  |-  ( ( 1st `  u ) H ( 2nd `  u
) )  =  ( H `  <. ( 1st `  u ) ,  ( 2nd `  u
) >. )
55 df-ov 6089 . . . . . . . . 9  |-  ( ( 1st `  u ) ( Hom  `  D
) ( 2nd `  u
) )  =  ( ( Hom  `  D
) `  <. ( 1st `  u ) ,  ( 2nd `  u )
>. )
5653, 54, 553eqtr3g 2493 . . . . . . . 8  |-  ( (
ph  /\  u  e.  ( B  X.  B
)  /\  z  e.  B )  ->  ( H `  <. ( 1st `  u ) ,  ( 2nd `  u )
>. )  =  (
( Hom  `  D ) `
 <. ( 1st `  u
) ,  ( 2nd `  u ) >. )
)
57 1st2nd2 6608 . . . . . . . . . 10  |-  ( u  e.  ( B  X.  B )  ->  u  =  <. ( 1st `  u
) ,  ( 2nd `  u ) >. )
58573ad2ant2 1010 . . . . . . . . 9  |-  ( (
ph  /\  u  e.  ( B  X.  B
)  /\  z  e.  B )  ->  u  =  <. ( 1st `  u
) ,  ( 2nd `  u ) >. )
5958fveq2d 5690 . . . . . . . 8  |-  ( (
ph  /\  u  e.  ( B  X.  B
)  /\  z  e.  B )  ->  ( H `  u )  =  ( H `  <. ( 1st `  u
) ,  ( 2nd `  u ) >. )
)
6058fveq2d 5690 . . . . . . . 8  |-  ( (
ph  /\  u  e.  ( B  X.  B
)  /\  z  e.  B )  ->  (
( Hom  `  D ) `
 u )  =  ( ( Hom  `  D
) `  <. ( 1st `  u ) ,  ( 2nd `  u )
>. ) )
6156, 59, 603eqtr4d 2480 . . . . . . 7  |-  ( (
ph  /\  u  e.  ( B  X.  B
)  /\  z  e.  B )  ->  ( H `  u )  =  ( ( Hom  `  D ) `  u
) )
62 eqidd 2439 . . . . . . 7  |-  ( (
ph  /\  u  e.  ( B  X.  B
)  /\  z  e.  B )  ->  (
g ( u  .xb  z ) f )  =  ( g ( u  .xb  z )
f ) )
6349, 61, 62mpt2eq123dv 6143 . . . . . 6  |-  ( (
ph  /\  u  e.  ( B  X.  B
)  /\  z  e.  B )  ->  (
g  e.  ( ( 2nd `  u ) H z ) ,  f  e.  ( H `
 u )  |->  ( g ( u  .xb  z ) f ) )  =  ( g  e.  ( ( 2nd `  u ) ( Hom  `  D ) z ) ,  f  e.  ( ( Hom  `  D
) `  u )  |->  ( g ( u 
.xb  z ) f ) ) )
6463mpt2eq3dva 6145 . . . . 5  |-  ( ph  ->  ( u  e.  ( B  X.  B ) ,  z  e.  B  |->  ( g  e.  ( ( 2nd `  u
) H z ) ,  f  e.  ( H `  u ) 
|->  ( g ( u 
.xb  z ) f ) ) )  =  ( u  e.  ( B  X.  B ) ,  z  e.  B  |->  ( g  e.  ( ( 2nd `  u
) ( Hom  `  D
) z ) ,  f  e.  ( ( Hom  `  D ) `  u )  |->  ( g ( u  .xb  z
) f ) ) ) )
65 comfeq.4 . . . . . . 7  |-  ( ph  ->  B  =  ( Base `  D ) )
6665, 65xpeq12d 4860 . . . . . 6  |-  ( ph  ->  ( B  X.  B
)  =  ( (
Base `  D )  X.  ( Base `  D
) ) )
67 eqidd 2439 . . . . . 6  |-  ( ph  ->  ( g  e.  ( ( 2nd `  u
) ( Hom  `  D
) z ) ,  f  e.  ( ( Hom  `  D ) `  u )  |->  ( g ( u  .xb  z
) f ) )  =  ( g  e.  ( ( 2nd `  u
) ( Hom  `  D
) z ) ,  f  e.  ( ( Hom  `  D ) `  u )  |->  ( g ( u  .xb  z
) f ) ) )
6866, 65, 67mpt2eq123dv 6143 . . . . 5  |-  ( ph  ->  ( u  e.  ( B  X.  B ) ,  z  e.  B  |->  ( g  e.  ( ( 2nd `  u
) ( Hom  `  D
) z ) ,  f  e.  ( ( Hom  `  D ) `  u )  |->  ( g ( u  .xb  z
) f ) ) )  =  ( u  e.  ( ( Base `  D )  X.  ( Base `  D ) ) ,  z  e.  (
Base `  D )  |->  ( g  e.  ( ( 2nd `  u
) ( Hom  `  D
) z ) ,  f  e.  ( ( Hom  `  D ) `  u )  |->  ( g ( u  .xb  z
) f ) ) ) )
6964, 68eqtrd 2470 . . . 4  |-  ( ph  ->  ( u  e.  ( B  X.  B ) ,  z  e.  B  |->  ( g  e.  ( ( 2nd `  u
) H z ) ,  f  e.  ( H `  u ) 
|->  ( g ( u 
.xb  z ) f ) ) )  =  ( u  e.  ( ( Base `  D
)  X.  ( Base `  D ) ) ,  z  e.  ( Base `  D )  |->  ( g  e.  ( ( 2nd `  u ) ( Hom  `  D ) z ) ,  f  e.  ( ( Hom  `  D
) `  u )  |->  ( g ( u 
.xb  z ) f ) ) ) )
70 eqid 2438 . . . . 5  |-  (compf `  D
)  =  (compf `  D
)
71 eqid 2438 . . . . 5  |-  ( Base `  D )  =  (
Base `  D )
72 comfeq.2 . . . . 5  |-  .xb  =  (comp `  D )
7370, 71, 40, 72comfffval 14629 . . . 4  |-  (compf `  D
)  =  ( u  e.  ( ( Base `  D )  X.  ( Base `  D ) ) ,  z  e.  (
Base `  D )  |->  ( g  e.  ( ( 2nd `  u
) ( Hom  `  D
) z ) ,  f  e.  ( ( Hom  `  D ) `  u )  |->  ( g ( u  .xb  z
) f ) ) )
7469, 73syl6eqr 2488 . . 3  |-  ( ph  ->  ( u  e.  ( B  X.  B ) ,  z  e.  B  |->  ( g  e.  ( ( 2nd `  u
) H z ) ,  f  e.  ( H `  u ) 
|->  ( g ( u 
.xb  z ) f ) ) )  =  (compf `  D ) )
7539, 74eqeq12d 2452 . 2  |-  ( ph  ->  ( ( u  e.  ( B  X.  B
) ,  z  e.  B  |->  ( g  e.  ( ( 2nd `  u
) H z ) ,  f  e.  ( H `  u ) 
|->  ( g ( u 
.x.  z ) f ) ) )  =  ( u  e.  ( B  X.  B ) ,  z  e.  B  |->  ( g  e.  ( ( 2nd `  u
) H z ) ,  f  e.  ( H `  u ) 
|->  ( g ( u 
.xb  z ) f ) ) )  <->  (compf `  C )  =  (compf `  D ) ) )
7629, 75syl5rbbr 260 1  |-  ( ph  ->  ( (compf `  C )  =  (compf `  D )  <->  A. x  e.  B  A. y  e.  B  A. z  e.  B  A. f  e.  ( x H y ) A. g  e.  ( y H z ) ( g (
<. x ,  y >.  .x.  z ) f )  =  ( g (
<. x ,  y >.  .xb  z ) f ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ w3a 965    = wceq 1369    e. wcel 1756   A.wral 2710   _Vcvv 2967   <.cop 3878    X. cxp 4833   ` cfv 5413  (class class class)co 6086    e. cmpt2 6088   1stc1st 6570   2ndc2nd 6571   Basecbs 14166   Hom chom 14241  compcco 14242   Hom f chomf 14596  compfccomf 14597
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-reu 2717  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-1st 6572  df-2nd 6573  df-homf 14600  df-comf 14601
This theorem is referenced by:  comfeqd  14638  2oppccomf  14656  oppccomfpropd  14658  resssetc  14952  resscatc  14965
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