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Theorem comfval 13696
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 13695 . 2  |-  ( ph  ->  ( <. X ,  Y >. O Z )  =  ( g  e.  ( Y H Z ) ,  f  e.  ( X H Y ) 
|->  ( g ( <. X ,  Y >.  .x. 
Z ) f ) ) )
9 oveq12 5951 . . 3  |-  ( ( g  =  G  /\  f  =  F )  ->  ( g ( <. X ,  Y >.  .x. 
Z ) f )  =  ( G (
<. X ,  Y >.  .x. 
Z ) F ) )
109adantl 452 . 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 5967 . . 3  |-  ( G ( <. X ,  Y >.  .x.  Z ) F )  e.  _V
1413a1i 10 . 2  |-  ( ph  ->  ( G ( <. X ,  Y >.  .x. 
Z ) F )  e.  _V )
158, 10, 11, 12, 14ovmpt2d 6059 1  |-  ( ph  ->  ( G ( <. X ,  Y >. O Z ) F )  =  ( G (
<. X ,  Y >.  .x. 
Z ) F ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1642    e. wcel 1710   _Vcvv 2864   <.cop 3719   ` cfv 5334  (class class class)co 5942   Basecbs 13239    Hom chom 13310  compcco 13311  compfccomf 13662
This theorem is referenced by:  comfval2  13699  comfeqval  13704
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4210  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3907  df-iun 3986  df-br 4103  df-opab 4157  df-mpt 4158  df-id 4388  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-ov 5945  df-oprab 5946  df-mpt2 5947  df-1st 6206  df-2nd 6207  df-comf 13666
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