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Theorem comfeqd 13860
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 6037 . . . . . . . 8  |-  ( ph  ->  ( <. x ,  y
>. (comp `  C )
z )  =  (
<. x ,  y >.
(comp `  D )
z ) )
32oveqd 6037 . . . . . . 7  |-  ( ph  ->  ( g ( <.
x ,  y >.
(comp `  C )
z ) f )  =  ( g (
<. x ,  y >.
(comp `  D )
z ) f ) )
43ralrimivw 2733 . . . . . 6  |-  ( ph  ->  A. g  e.  ( y (  Hom  `  C
) z ) ( g ( <. x ,  y >. (comp `  C ) z ) f )  =  ( g ( <. x ,  y >. (comp `  D ) z ) f ) )
54ralrimivw 2733 . . . . 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 2733 . . . 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 2733 . . 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 2733 . 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 2387 . . 3  |-  (comp `  C )  =  (comp `  C )
10 eqid 2387 . . 3  |-  (comp `  D )  =  (comp `  D )
11 eqid 2387 . . 3  |-  (  Hom  `  C )  =  (  Hom  `  C )
12 eqidd 2388 . . 3  |-  ( ph  ->  ( Base `  C
)  =  ( Base `  C ) )
13 comfeqd.2 . . . 4  |-  ( ph  ->  (  Homf 
`  C )  =  (  Homf 
`  D ) )
1413homfeqbas 13849 . . 3  |-  ( ph  ->  ( Base `  C
)  =  ( Base `  D ) )
159, 10, 11, 12, 14, 13comfeq 13859 . 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 224 1  |-  ( ph  ->  (compf `  C )  =  (compf `  D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649   A.wral 2649   <.cop 3760   ` cfv 5394  (class class class)co 6020   Basecbs 13396    Hom chom 13467  compcco 13468    Homf chomf 13818  compfccomf 13819
This theorem is referenced by:  fullresc  13975
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-reu 2656  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-homf 13822  df-comf 13823
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