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Theorem comfeq 14961
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 6308 . . . . . 6  |-  ( ( 2nd `  u ) H z )  e. 
_V
2 fvex 5875 . . . . . 6  |-  ( H `
 u )  e. 
_V
31, 2mpt2ex 6860 . . . . 5  |-  ( g  e.  ( ( 2nd `  u ) H z ) ,  f  e.  ( H `  u
)  |->  ( g ( u  .x.  z ) f ) )  e. 
_V
43rgen2w 2826 . . . 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 6394 . . . 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 3116 . . . . . . . . 9  |-  x  e. 
_V
8 vex 3116 . . . . . . . . 9  |-  y  e. 
_V
97, 8op2ndd 6795 . . . . . . . 8  |-  ( u  =  <. x ,  y
>.  ->  ( 2nd `  u
)  =  y )
109oveq1d 6298 . . . . . . 7  |-  ( u  =  <. x ,  y
>.  ->  ( ( 2nd `  u ) H z )  =  ( y H z ) )
11 fveq2 5865 . . . . . . . . 9  |-  ( u  =  <. x ,  y
>.  ->  ( H `  u )  =  ( H `  <. x ,  y >. )
)
12 df-ov 6286 . . . . . . . . 9  |-  ( x H y )  =  ( H `  <. x ,  y >. )
1311, 12syl6eqr 2526 . . . . . . . 8  |-  ( u  =  <. x ,  y
>.  ->  ( H `  u )  =  ( x H y ) )
14 oveq1 6290 . . . . . . . . . 10  |-  ( u  =  <. x ,  y
>.  ->  ( u  .x.  z )  =  (
<. x ,  y >.  .x.  z ) )
1514oveqd 6300 . . . . . . . . 9  |-  ( u  =  <. x ,  y
>.  ->  ( g ( u  .x.  z ) f )  =  ( g ( <. x ,  y >.  .x.  z
) f ) )
16 oveq1 6290 . . . . . . . . . 10  |-  ( u  =  <. x ,  y
>.  ->  ( u  .xb  z )  =  (
<. x ,  y >.  .xb  z ) )
1716oveqd 6300 . . . . . . . . 9  |-  ( u  =  <. x ,  y
>.  ->  ( g ( u  .xb  z )
f )  =  ( g ( <. x ,  y >.  .xb  z
) f ) )
1815, 17eqeq12d 2489 . . . . . . . 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 3072 . . . . . . 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 3072 . . . . . 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 6308 . . . . . . . 8  |-  ( g ( u  .x.  z
) f )  e. 
_V
2221rgen2w 2826 . . . . . . 7  |-  A. g  e.  ( ( 2nd `  u
) H z ) A. f  e.  ( H `  u ) ( g ( u 
.x.  z ) f )  e.  _V
23 mpt22eqb 6394 . . . . . . 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 3022 . . . . . 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 2903 . . . 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 5143 . . 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 5024 . . . . 5  |-  ( ph  ->  ( B  X.  B
)  =  ( (
Base `  C )  X.  ( Base `  C
) ) )
32 eqidd 2468 . . . . 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 6342 . . . 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 2467 . . . . 5  |-  (compf `  C
)  =  (compf `  C
)
35 eqid 2467 . . . . 5  |-  ( Base `  C )  =  (
Base `  C )
36 comfeq.h . . . . 5  |-  H  =  ( Hom  `  C
)
37 comfeq.1 . . . . 5  |-  .x.  =  (comp `  C )
3834, 35, 36, 37comfffval 14953 . . . 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 2526 . . 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 2467 . . . . . . . 8  |-  ( Hom  `  D )  =  ( Hom  `  D )
41 comfeq.5 . . . . . . . . 9  |-  ( ph  ->  ( Hom f  `  C )  =  ( Hom f  `  D ) )
42413ad2ant1 1017 . . . . . . . 8  |-  ( (
ph  /\  u  e.  ( B  X.  B
)  /\  z  e.  B )  ->  ( Hom f  `  C )  =  ( Hom f  `  D ) )
43 xp2nd 6815 . . . . . . . . . 10  |-  ( u  e.  ( B  X.  B )  ->  ( 2nd `  u )  e.  B )
44433ad2ant2 1018 . . . . . . . . 9  |-  ( (
ph  /\  u  e.  ( B  X.  B
)  /\  z  e.  B )  ->  ( 2nd `  u )  e.  B )
45303ad2ant1 1017 . . . . . . . . 9  |-  ( (
ph  /\  u  e.  ( B  X.  B
)  /\  z  e.  B )  ->  B  =  ( Base `  C
) )
4644, 45eleqtrd 2557 . . . . . . . 8  |-  ( (
ph  /\  u  e.  ( B  X.  B
)  /\  z  e.  B )  ->  ( 2nd `  u )  e.  ( Base `  C
) )
47 simp3 998 . . . . . . . . 9  |-  ( (
ph  /\  u  e.  ( B  X.  B
)  /\  z  e.  B )  ->  z  e.  B )
4847, 45eleqtrd 2557 . . . . . . . 8  |-  ( (
ph  /\  u  e.  ( B  X.  B
)  /\  z  e.  B )  ->  z  e.  ( Base `  C
) )
4935, 36, 40, 42, 46, 48homfeqval 14952 . . . . . . 7  |-  ( (
ph  /\  u  e.  ( B  X.  B
)  /\  z  e.  B )  ->  (
( 2nd `  u
) H z )  =  ( ( 2nd `  u ) ( Hom  `  D ) z ) )
50 xp1st 6814 . . . . . . . . . . . 12  |-  ( u  e.  ( B  X.  B )  ->  ( 1st `  u )  e.  B )
51503ad2ant2 1018 . . . . . . . . . . 11  |-  ( (
ph  /\  u  e.  ( B  X.  B
)  /\  z  e.  B )  ->  ( 1st `  u )  e.  B )
5251, 45eleqtrd 2557 . . . . . . . . . 10  |-  ( (
ph  /\  u  e.  ( B  X.  B
)  /\  z  e.  B )  ->  ( 1st `  u )  e.  ( Base `  C
) )
5335, 36, 40, 42, 52, 46homfeqval 14952 . . . . . . . . 9  |-  ( (
ph  /\  u  e.  ( B  X.  B
)  /\  z  e.  B )  ->  (
( 1st `  u
) H ( 2nd `  u ) )  =  ( ( 1st `  u
) ( Hom  `  D
) ( 2nd `  u
) ) )
54 df-ov 6286 . . . . . . . . 9  |-  ( ( 1st `  u ) H ( 2nd `  u
) )  =  ( H `  <. ( 1st `  u ) ,  ( 2nd `  u
) >. )
55 df-ov 6286 . . . . . . . . 9  |-  ( ( 1st `  u ) ( Hom  `  D
) ( 2nd `  u
) )  =  ( ( Hom  `  D
) `  <. ( 1st `  u ) ,  ( 2nd `  u )
>. )
5653, 54, 553eqtr3g 2531 . . . . . . . 8  |-  ( (
ph  /\  u  e.  ( B  X.  B
)  /\  z  e.  B )  ->  ( H `  <. ( 1st `  u ) ,  ( 2nd `  u )
>. )  =  (
( Hom  `  D ) `
 <. ( 1st `  u
) ,  ( 2nd `  u ) >. )
)
57 1st2nd2 6821 . . . . . . . . . 10  |-  ( u  e.  ( B  X.  B )  ->  u  =  <. ( 1st `  u
) ,  ( 2nd `  u ) >. )
58573ad2ant2 1018 . . . . . . . . 9  |-  ( (
ph  /\  u  e.  ( B  X.  B
)  /\  z  e.  B )  ->  u  =  <. ( 1st `  u
) ,  ( 2nd `  u ) >. )
5958fveq2d 5869 . . . . . . . 8  |-  ( (
ph  /\  u  e.  ( B  X.  B
)  /\  z  e.  B )  ->  ( H `  u )  =  ( H `  <. ( 1st `  u
) ,  ( 2nd `  u ) >. )
)
6058fveq2d 5869 . . . . . . . 8  |-  ( (
ph  /\  u  e.  ( B  X.  B
)  /\  z  e.  B )  ->  (
( Hom  `  D ) `
 u )  =  ( ( Hom  `  D
) `  <. ( 1st `  u ) ,  ( 2nd `  u )
>. ) )
6156, 59, 603eqtr4d 2518 . . . . . . 7  |-  ( (
ph  /\  u  e.  ( B  X.  B
)  /\  z  e.  B )  ->  ( H `  u )  =  ( ( Hom  `  D ) `  u
) )
62 eqidd 2468 . . . . . . 7  |-  ( (
ph  /\  u  e.  ( B  X.  B
)  /\  z  e.  B )  ->  (
g ( u  .xb  z ) f )  =  ( g ( u  .xb  z )
f ) )
6349, 61, 62mpt2eq123dv 6342 . . . . . 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 6344 . . . . 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 5024 . . . . . 6  |-  ( ph  ->  ( B  X.  B
)  =  ( (
Base `  D )  X.  ( Base `  D
) ) )
67 eqidd 2468 . . . . . 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 6342 . . . . 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 2508 . . . 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 2467 . . . . 5  |-  (compf `  D
)  =  (compf `  D
)
71 eqid 2467 . . . . 5  |-  ( Base `  D )  =  (
Base `  D )
72 comfeq.2 . . . . 5  |-  .xb  =  (comp `  D )
7370, 71, 40, 72comfffval 14953 . . . 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 2526 . . 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 2489 . 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 973    = wceq 1379    e. wcel 1767   A.wral 2814   _Vcvv 3113   <.cop 4033    X. cxp 4997   ` cfv 5587  (class class class)co 6283    |-> cmpt2 6285   1stc1st 6782   2ndc2nd 6783   Basecbs 14489   Hom chom 14565  compcco 14566   Hom f chomf 14920  compfccomf 14921
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6575
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5550  df-fun 5589  df-fn 5590  df-f 5591  df-f1 5592  df-fo 5593  df-f1o 5594  df-fv 5595  df-ov 6286  df-oprab 6287  df-mpt2 6288  df-1st 6784  df-2nd 6785  df-homf 14924  df-comf 14925
This theorem is referenced by:  comfeqd  14962  2oppccomf  14980  oppccomfpropd  14982  resssetc  15276  resscatc  15289
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