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Theorem comfeq 14767
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 6228 . . . . . 6  |-  ( ( 2nd `  u ) H z )  e. 
_V
2 fvex 5812 . . . . . 6  |-  ( H `
 u )  e. 
_V
31, 2mpt2ex 6763 . . . . 5  |-  ( g  e.  ( ( 2nd `  u ) H z ) ,  f  e.  ( H `  u
)  |->  ( g ( u  .x.  z ) f ) )  e. 
_V
43rgen2w 2902 . . . 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 6312 . . . 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 3081 . . . . . . . . 9  |-  x  e. 
_V
8 vex 3081 . . . . . . . . 9  |-  y  e. 
_V
97, 8op2ndd 6701 . . . . . . . 8  |-  ( u  =  <. x ,  y
>.  ->  ( 2nd `  u
)  =  y )
109oveq1d 6218 . . . . . . 7  |-  ( u  =  <. x ,  y
>.  ->  ( ( 2nd `  u ) H z )  =  ( y H z ) )
11 fveq2 5802 . . . . . . . . 9  |-  ( u  =  <. x ,  y
>.  ->  ( H `  u )  =  ( H `  <. x ,  y >. )
)
12 df-ov 6206 . . . . . . . . 9  |-  ( x H y )  =  ( H `  <. x ,  y >. )
1311, 12syl6eqr 2513 . . . . . . . 8  |-  ( u  =  <. x ,  y
>.  ->  ( H `  u )  =  ( x H y ) )
14 oveq1 6210 . . . . . . . . . 10  |-  ( u  =  <. x ,  y
>.  ->  ( u  .x.  z )  =  (
<. x ,  y >.  .x.  z ) )
1514oveqd 6220 . . . . . . . . 9  |-  ( u  =  <. x ,  y
>.  ->  ( g ( u  .x.  z ) f )  =  ( g ( <. x ,  y >.  .x.  z
) f ) )
16 oveq1 6210 . . . . . . . . . 10  |-  ( u  =  <. x ,  y
>.  ->  ( u  .xb  z )  =  (
<. x ,  y >.  .xb  z ) )
1716oveqd 6220 . . . . . . . . 9  |-  ( u  =  <. x ,  y
>.  ->  ( g ( u  .xb  z )
f )  =  ( g ( <. x ,  y >.  .xb  z
) f ) )
1815, 17eqeq12d 2476 . . . . . . . 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 3037 . . . . . . 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 3037 . . . . . 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 6228 . . . . . . . 8  |-  ( g ( u  .x.  z
) f )  e. 
_V
2221rgen2w 2902 . . . . . . 7  |-  A. g  e.  ( ( 2nd `  u
) H z ) A. f  e.  ( H `  u ) ( g ( u 
.x.  z ) f )  e.  _V
23 mpt22eqb 6312 . . . . . . 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 2987 . . . . . 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 2846 . . . 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 5092 . . 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 4976 . . . . 5  |-  ( ph  ->  ( B  X.  B
)  =  ( (
Base `  C )  X.  ( Base `  C
) ) )
32 eqidd 2455 . . . . 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 6260 . . . 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 2454 . . . . 5  |-  (compf `  C
)  =  (compf `  C
)
35 eqid 2454 . . . . 5  |-  ( Base `  C )  =  (
Base `  C )
36 comfeq.h . . . . 5  |-  H  =  ( Hom  `  C
)
37 comfeq.1 . . . . 5  |-  .x.  =  (comp `  C )
3834, 35, 36, 37comfffval 14759 . . . 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 2513 . . 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 2454 . . . . . . . 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 6720 . . . . . . . . . 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 2544 . . . . . . . 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 2544 . . . . . . . 8  |-  ( (
ph  /\  u  e.  ( B  X.  B
)  /\  z  e.  B )  ->  z  e.  ( Base `  C
) )
4935, 36, 40, 42, 46, 48homfeqval 14758 . . . . . . 7  |-  ( (
ph  /\  u  e.  ( B  X.  B
)  /\  z  e.  B )  ->  (
( 2nd `  u
) H z )  =  ( ( 2nd `  u ) ( Hom  `  D ) z ) )
50 xp1st 6719 . . . . . . . . . . . 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 2544 . . . . . . . . . 10  |-  ( (
ph  /\  u  e.  ( B  X.  B
)  /\  z  e.  B )  ->  ( 1st `  u )  e.  ( Base `  C
) )
5335, 36, 40, 42, 52, 46homfeqval 14758 . . . . . . . . 9  |-  ( (
ph  /\  u  e.  ( B  X.  B
)  /\  z  e.  B )  ->  (
( 1st `  u
) H ( 2nd `  u ) )  =  ( ( 1st `  u
) ( Hom  `  D
) ( 2nd `  u
) ) )
54 df-ov 6206 . . . . . . . . 9  |-  ( ( 1st `  u ) H ( 2nd `  u
) )  =  ( H `  <. ( 1st `  u ) ,  ( 2nd `  u
) >. )
55 df-ov 6206 . . . . . . . . 9  |-  ( ( 1st `  u ) ( Hom  `  D
) ( 2nd `  u
) )  =  ( ( Hom  `  D
) `  <. ( 1st `  u ) ,  ( 2nd `  u )
>. )
5653, 54, 553eqtr3g 2518 . . . . . . . 8  |-  ( (
ph  /\  u  e.  ( B  X.  B
)  /\  z  e.  B )  ->  ( H `  <. ( 1st `  u ) ,  ( 2nd `  u )
>. )  =  (
( Hom  `  D ) `
 <. ( 1st `  u
) ,  ( 2nd `  u ) >. )
)
57 1st2nd2 6726 . . . . . . . . . 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 5806 . . . . . . . 8  |-  ( (
ph  /\  u  e.  ( B  X.  B
)  /\  z  e.  B )  ->  ( H `  u )  =  ( H `  <. ( 1st `  u
) ,  ( 2nd `  u ) >. )
)
6058fveq2d 5806 . . . . . . . 8  |-  ( (
ph  /\  u  e.  ( B  X.  B
)  /\  z  e.  B )  ->  (
( Hom  `  D ) `
 u )  =  ( ( Hom  `  D
) `  <. ( 1st `  u ) ,  ( 2nd `  u )
>. ) )
6156, 59, 603eqtr4d 2505 . . . . . . 7  |-  ( (
ph  /\  u  e.  ( B  X.  B
)  /\  z  e.  B )  ->  ( H `  u )  =  ( ( Hom  `  D ) `  u
) )
62 eqidd 2455 . . . . . . 7  |-  ( (
ph  /\  u  e.  ( B  X.  B
)  /\  z  e.  B )  ->  (
g ( u  .xb  z ) f )  =  ( g ( u  .xb  z )
f ) )
6349, 61, 62mpt2eq123dv 6260 . . . . . 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 6262 . . . . 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 4976 . . . . . 6  |-  ( ph  ->  ( B  X.  B
)  =  ( (
Base `  D )  X.  ( Base `  D
) ) )
67 eqidd 2455 . . . . . 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 6260 . . . . 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 2495 . . . 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 2454 . . . . 5  |-  (compf `  D
)  =  (compf `  D
)
71 eqid 2454 . . . . 5  |-  ( Base `  D )  =  (
Base `  D )
72 comfeq.2 . . . . 5  |-  .xb  =  (comp `  D )
7370, 71, 40, 72comfffval 14759 . . . 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 2513 . . 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 2476 . 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 1370    e. wcel 1758   A.wral 2799   _Vcvv 3078   <.cop 3994    X. cxp 4949   ` cfv 5529  (class class class)co 6203    |-> cmpt2 6205   1stc1st 6688   2ndc2nd 6689   Basecbs 14295   Hom chom 14371  compcco 14372   Hom f chomf 14726  compfccomf 14727
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-1st 6690  df-2nd 6691  df-homf 14730  df-comf 14731
This theorem is referenced by:  comfeqd  14768  2oppccomf  14786  oppccomfpropd  14788  resssetc  15082  resscatc  15095
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