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Theorem comfeqd 14665
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  ->  ( Hom f  `  C )  =  ( Hom f  `  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 6127 . . . . . . . 8  |-  ( ph  ->  ( <. x ,  y
>. (comp `  C )
z )  =  (
<. x ,  y >.
(comp `  D )
z ) )
32oveqd 6127 . . . . . . 7  |-  ( ph  ->  ( g ( <.
x ,  y >.
(comp `  C )
z ) f )  =  ( g (
<. x ,  y >.
(comp `  D )
z ) f ) )
43ralrimivw 2819 . . . . . 6  |-  ( ph  ->  A. g  e.  ( y ( Hom  `  C
) z ) ( g ( <. x ,  y >. (comp `  C ) z ) f )  =  ( g ( <. x ,  y >. (comp `  D ) z ) f ) )
54ralrimivw 2819 . . . . 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 2819 . . . 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 2819 . . 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 2819 . 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 2443 . . 3  |-  (comp `  C )  =  (comp `  C )
10 eqid 2443 . . 3  |-  (comp `  D )  =  (comp `  D )
11 eqid 2443 . . 3  |-  ( Hom  `  C )  =  ( Hom  `  C )
12 eqidd 2444 . . 3  |-  ( ph  ->  ( Base `  C
)  =  ( Base `  C ) )
13 comfeqd.2 . . . 4  |-  ( ph  ->  ( Hom f  `  C )  =  ( Hom f  `  D ) )
1413homfeqbas 14654 . . 3  |-  ( ph  ->  ( Base `  C
)  =  ( Base `  D ) )
159, 10, 11, 12, 14, 13comfeq 14664 . 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 232 1  |-  ( ph  ->  (compf `  C )  =  (compf `  D ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1369   A.wral 2734   <.cop 3902   ` cfv 5437  (class class class)co 6110   Basecbs 14193   Hom chom 14268  compcco 14269   Hom f chomf 14623  compfccomf 14624
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 2423  ax-rep 4422  ax-sep 4432  ax-nul 4440  ax-pow 4489  ax-pr 4550  ax-un 6391
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 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2739  df-rex 2740  df-reu 2741  df-rab 2743  df-v 2993  df-sbc 3206  df-csb 3308  df-dif 3350  df-un 3352  df-in 3354  df-ss 3361  df-nul 3657  df-if 3811  df-pw 3881  df-sn 3897  df-pr 3899  df-op 3903  df-uni 4111  df-iun 4192  df-br 4312  df-opab 4370  df-mpt 4371  df-id 4655  df-xp 4865  df-rel 4866  df-cnv 4867  df-co 4868  df-dm 4869  df-rn 4870  df-res 4871  df-ima 4872  df-iota 5400  df-fun 5439  df-fn 5440  df-f 5441  df-f1 5442  df-fo 5443  df-f1o 5444  df-fv 5445  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-1st 6596  df-2nd 6597  df-homf 14627  df-comf 14628
This theorem is referenced by:  fullresc  14780
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