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Theorem comfeqd 13925
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 6090 . . . . . . . 8  |-  ( ph  ->  ( <. x ,  y
>. (comp `  C )
z )  =  (
<. x ,  y >.
(comp `  D )
z ) )
32oveqd 6090 . . . . . . 7  |-  ( ph  ->  ( g ( <.
x ,  y >.
(comp `  C )
z ) f )  =  ( g (
<. x ,  y >.
(comp `  D )
z ) f ) )
43ralrimivw 2782 . . . . . 6  |-  ( ph  ->  A. g  e.  ( y (  Hom  `  C
) z ) ( g ( <. x ,  y >. (comp `  C ) z ) f )  =  ( g ( <. x ,  y >. (comp `  D ) z ) f ) )
54ralrimivw 2782 . . . . 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 2782 . . . 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 2782 . . 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 2782 . 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 2435 . . 3  |-  (comp `  C )  =  (comp `  C )
10 eqid 2435 . . 3  |-  (comp `  D )  =  (comp `  D )
11 eqid 2435 . . 3  |-  (  Hom  `  C )  =  (  Hom  `  C )
12 eqidd 2436 . . 3  |-  ( ph  ->  ( Base `  C
)  =  ( Base `  C ) )
13 comfeqd.2 . . . 4  |-  ( ph  ->  (  Homf 
`  C )  =  (  Homf 
`  D ) )
1413homfeqbas 13914 . . 3  |-  ( ph  ->  ( Base `  C
)  =  ( Base `  D ) )
159, 10, 11, 12, 14, 13comfeq 13924 . 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 1652   A.wral 2697   <.cop 3809   ` cfv 5446  (class class class)co 6073   Basecbs 13461    Hom chom 13532  compcco 13533    Homf chomf 13883  compfccomf 13884
This theorem is referenced by:  fullresc  14040
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-homf 13887  df-comf 13888
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