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Theorem comfeqd 13626
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
comfeqd.1  |-  ( ph  ->  (comp `  C )  =  (comp `  D )
)
comfeqd.2  |-  ( ph  ->  (  Homf 
`  C )  =  (  Homf 
`  D ) )
Assertion
Ref Expression
comfeqd  |-  ( ph  ->  (compf `  C )  =  (compf `  D ) )

Proof of Theorem comfeqd
Dummy variables  f 
g  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 comfeqd.1 . . . . . . . . 9  |-  ( ph  ->  (comp `  C )  =  (comp `  D )
)
21oveqd 5891 . . . . . . . 8  |-  ( ph  ->  ( <. x ,  y
>. (comp `  C )
z )  =  (
<. x ,  y >.
(comp `  D )
z ) )
32oveqd 5891 . . . . . . 7  |-  ( ph  ->  ( g ( <.
x ,  y >.
(comp `  C )
z ) f )  =  ( g (
<. x ,  y >.
(comp `  D )
z ) f ) )
43ralrimivw 2640 . . . . . 6  |-  ( ph  ->  A. g  e.  ( y (  Hom  `  C
) z ) ( g ( <. x ,  y >. (comp `  C ) z ) f )  =  ( g ( <. x ,  y >. (comp `  D ) z ) f ) )
54ralrimivw 2640 . . . . 5  |-  ( ph  ->  A. f  e.  ( x (  Hom  `  C
) y ) A. g  e.  ( y
(  Hom  `  C ) z ) ( g ( <. x ,  y
>. (comp `  C )
z ) f )  =  ( g (
<. x ,  y >.
(comp `  D )
z ) f ) )
65ralrimivw 2640 . . . 4  |-  ( ph  ->  A. z  e.  (
Base `  C ) A. f  e.  (
x (  Hom  `  C
) y ) A. g  e.  ( y
(  Hom  `  C ) z ) ( g ( <. x ,  y
>. (comp `  C )
z ) f )  =  ( g (
<. x ,  y >.
(comp `  D )
z ) f ) )
76ralrimivw 2640 . . 3  |-  ( ph  ->  A. y  e.  (
Base `  C ) A. z  e.  ( Base `  C ) A. f  e.  ( x
(  Hom  `  C ) y ) A. g  e.  ( y (  Hom  `  C ) z ) ( g ( <.
x ,  y >.
(comp `  C )
z ) f )  =  ( g (
<. x ,  y >.
(comp `  D )
z ) f ) )
87ralrimivw 2640 . 2  |-  ( ph  ->  A. x  e.  (
Base `  C ) A. y  e.  ( Base `  C ) A. z  e.  ( Base `  C ) A. f  e.  ( x (  Hom  `  C ) y ) A. g  e.  ( y (  Hom  `  C
) z ) ( g ( <. x ,  y >. (comp `  C ) z ) f )  =  ( g ( <. x ,  y >. (comp `  D ) z ) f ) )
9 eqid 2296 . . 3  |-  (comp `  C )  =  (comp `  C )
10 eqid 2296 . . 3  |-  (comp `  D )  =  (comp `  D )
11 eqid 2296 . . 3  |-  (  Hom  `  C )  =  (  Hom  `  C )
12 eqidd 2297 . . 3  |-  ( ph  ->  ( Base `  C
)  =  ( Base `  C ) )
13 comfeqd.2 . . . 4  |-  ( ph  ->  (  Homf 
`  C )  =  (  Homf 
`  D ) )
1413homfeqbas 13615 . . 3  |-  ( ph  ->  ( Base `  C
)  =  ( Base `  D ) )
159, 10, 11, 12, 14, 13comfeq 13625 . 2  |-  ( ph  ->  ( (compf `  C )  =  (compf `  D )  <->  A. x  e.  ( Base `  C
) A. y  e.  ( Base `  C
) A. z  e.  ( Base `  C
) A. f  e.  ( x (  Hom  `  C ) y ) A. g  e.  ( y (  Hom  `  C
) z ) ( g ( <. x ,  y >. (comp `  C ) z ) f )  =  ( g ( <. x ,  y >. (comp `  D ) z ) f ) ) )
168, 15mpbird 223 1  |-  ( ph  ->  (compf `  C )  =  (compf `  D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632   A.wral 2556   <.cop 3656   ` cfv 5271  (class class class)co 5874   Basecbs 13164    Hom chom 13235  compcco 13236    Homf chomf 13584  compfccomf 13585
This theorem is referenced by:  fullresc  13741
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-homf 13588  df-comf 13589
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