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Theorem comfval 14635
Description: Value of the functionalized composition operation. (Contributed by Mario Carneiro, 4-Jan-2017.)
Hypotheses
Ref Expression
comfffval.o  |-  O  =  (compf `  C )
comfffval.b  |-  B  =  ( Base `  C
)
comfffval.h  |-  H  =  ( Hom  `  C
)
comfffval.x  |-  .x.  =  (comp `  C )
comffval.x  |-  ( ph  ->  X  e.  B )
comffval.y  |-  ( ph  ->  Y  e.  B )
comffval.z  |-  ( ph  ->  Z  e.  B )
comfval.f  |-  ( ph  ->  F  e.  ( X H Y ) )
comfval.g  |-  ( ph  ->  G  e.  ( Y H Z ) )
Assertion
Ref Expression
comfval  |-  ( ph  ->  ( G ( <. X ,  Y >. O Z ) F )  =  ( G (
<. X ,  Y >.  .x. 
Z ) F ) )

Proof of Theorem comfval
Dummy variables  f 
g are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 comfffval.o . . 3  |-  O  =  (compf `  C )
2 comfffval.b . . 3  |-  B  =  ( Base `  C
)
3 comfffval.h . . 3  |-  H  =  ( Hom  `  C
)
4 comfffval.x . . 3  |-  .x.  =  (comp `  C )
5 comffval.x . . 3  |-  ( ph  ->  X  e.  B )
6 comffval.y . . 3  |-  ( ph  ->  Y  e.  B )
7 comffval.z . . 3  |-  ( ph  ->  Z  e.  B )
81, 2, 3, 4, 5, 6, 7comffval 14634 . 2  |-  ( ph  ->  ( <. X ,  Y >. O Z )  =  ( g  e.  ( Y H Z ) ,  f  e.  ( X H Y ) 
|->  ( g ( <. X ,  Y >.  .x. 
Z ) f ) ) )
9 oveq12 6099 . . 3  |-  ( ( g  =  G  /\  f  =  F )  ->  ( g ( <. X ,  Y >.  .x. 
Z ) f )  =  ( G (
<. X ,  Y >.  .x. 
Z ) F ) )
109adantl 463 . 2  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  -> 
( g ( <. X ,  Y >.  .x. 
Z ) f )  =  ( G (
<. X ,  Y >.  .x. 
Z ) F ) )
11 comfval.g . 2  |-  ( ph  ->  G  e.  ( Y H Z ) )
12 comfval.f . 2  |-  ( ph  ->  F  e.  ( X H Y ) )
13 ovex 6115 . . 3  |-  ( G ( <. X ,  Y >.  .x.  Z ) F )  e.  _V
1413a1i 11 . 2  |-  ( ph  ->  ( G ( <. X ,  Y >.  .x. 
Z ) F )  e.  _V )
158, 10, 11, 12, 14ovmpt2d 6217 1  |-  ( ph  ->  ( G ( <. X ,  Y >. O Z ) F )  =  ( G (
<. X ,  Y >.  .x. 
Z ) F ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1364    e. wcel 1761   _Vcvv 2970   <.cop 3880   ` cfv 5415  (class class class)co 6090   Basecbs 14170   Hom chom 14245  compcco 14246  compfccomf 14601
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-1st 6576  df-2nd 6577  df-comf 14605
This theorem is referenced by:  comfval2  14638  comfeqval  14643
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