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Theorem comfval2 13606
Description: Value of the functionalized composition operation. (Contributed by Mario Carneiro, 4-Jan-2017.)
Hypotheses
Ref Expression
comfffval2.o  |-  O  =  (compf `  C )
comfffval2.b  |-  B  =  ( Base `  C
)
comfffval2.h  |-  H  =  (  Homf 
`  C )
comfffval2.x  |-  .x.  =  (comp `  C )
comffval2.x  |-  ( ph  ->  X  e.  B )
comffval2.y  |-  ( ph  ->  Y  e.  B )
comffval2.z  |-  ( ph  ->  Z  e.  B )
comfval2.f  |-  ( ph  ->  F  e.  ( X H Y ) )
comfval2.g  |-  ( ph  ->  G  e.  ( Y H Z ) )
Assertion
Ref Expression
comfval2  |-  ( ph  ->  ( G ( <. X ,  Y >. O Z ) F )  =  ( G (
<. X ,  Y >.  .x. 
Z ) F ) )

Proof of Theorem comfval2
StepHypRef Expression
1 comfffval2.o . 2  |-  O  =  (compf `  C )
2 comfffval2.b . 2  |-  B  =  ( Base `  C
)
3 eqid 2283 . 2  |-  (  Hom  `  C )  =  (  Hom  `  C )
4 comfffval2.x . 2  |-  .x.  =  (comp `  C )
5 comffval2.x . 2  |-  ( ph  ->  X  e.  B )
6 comffval2.y . 2  |-  ( ph  ->  Y  e.  B )
7 comffval2.z . 2  |-  ( ph  ->  Z  e.  B )
8 comfval2.f . . 3  |-  ( ph  ->  F  e.  ( X H Y ) )
9 comfffval2.h . . . 4  |-  H  =  (  Homf 
`  C )
109, 2, 3, 5, 6homfval 13595 . . 3  |-  ( ph  ->  ( X H Y )  =  ( X (  Hom  `  C
) Y ) )
118, 10eleqtrd 2359 . 2  |-  ( ph  ->  F  e.  ( X (  Hom  `  C
) Y ) )
12 comfval2.g . . 3  |-  ( ph  ->  G  e.  ( Y H Z ) )
139, 2, 3, 6, 7homfval 13595 . . 3  |-  ( ph  ->  ( Y H Z )  =  ( Y (  Hom  `  C
) Z ) )
1412, 13eleqtrd 2359 . 2  |-  ( ph  ->  G  e.  ( Y (  Hom  `  C
) Z ) )
151, 2, 3, 4, 5, 6, 7, 11, 14comfval 13603 1  |-  ( ph  ->  ( G ( <. X ,  Y >. O Z ) F )  =  ( G (
<. X ,  Y >.  .x. 
Z ) F ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   <.cop 3643   ` cfv 5255  (class class class)co 5858   Basecbs 13148    Hom chom 13219  compcco 13220    Homf chomf 13568  compfccomf 13569
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-homf 13572  df-comf 13573
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