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Theorem comffval2 13921
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 )
Assertion
Ref Expression
comffval2  |-  ( ph  ->  ( <. X ,  Y >. O Z )  =  ( g  e.  ( Y H Z ) ,  f  e.  ( X H Y ) 
|->  ( g ( <. X ,  Y >.  .x. 
Z ) f ) ) )
Distinct variable groups:    f, g, B    C, f, g    .x. , f,
g    f, X, g    f, Y, g    ph, f, g   
f, Z, g
Allowed substitution hints:    H( f, g)    O( f, g)

Proof of Theorem comffval2
StepHypRef Expression
1 comfffval2.o . . 3  |-  O  =  (compf `  C )
2 comfffval2.b . . 3  |-  B  =  ( Base `  C
)
3 eqid 2436 . . 3  |-  (  Hom  `  C )  =  (  Hom  `  C )
4 comfffval2.x . . 3  |-  .x.  =  (comp `  C )
5 comffval2.x . . 3  |-  ( ph  ->  X  e.  B )
6 comffval2.y . . 3  |-  ( ph  ->  Y  e.  B )
7 comffval2.z . . 3  |-  ( ph  ->  Z  e.  B )
81, 2, 3, 4, 5, 6, 7comffval 13918 . 2  |-  ( ph  ->  ( <. X ,  Y >. O Z )  =  ( g  e.  ( Y (  Hom  `  C
) Z ) ,  f  e.  ( X (  Hom  `  C
) Y )  |->  ( g ( <. X ,  Y >.  .x.  Z )
f ) ) )
9 comfffval2.h . . . 4  |-  H  =  (  Homf 
`  C )
109, 2, 3, 6, 7homfval 13911 . . 3  |-  ( ph  ->  ( Y H Z )  =  ( Y (  Hom  `  C
) Z ) )
119, 2, 3, 5, 6homfval 13911 . . 3  |-  ( ph  ->  ( X H Y )  =  ( X (  Hom  `  C
) Y ) )
12 eqidd 2437 . . 3  |-  ( ph  ->  ( g ( <. X ,  Y >.  .x. 
Z ) f )  =  ( g (
<. X ,  Y >.  .x. 
Z ) f ) )
1310, 11, 12mpt2eq123dv 6129 . 2  |-  ( ph  ->  ( g  e.  ( Y H Z ) ,  f  e.  ( X H Y ) 
|->  ( g ( <. X ,  Y >.  .x. 
Z ) f ) )  =  ( g  e.  ( Y (  Hom  `  C ) Z ) ,  f  e.  ( X (  Hom  `  C ) Y )  |->  ( g ( <. X ,  Y >.  .x.  Z ) f ) ) )
148, 13eqtr4d 2471 1  |-  ( ph  ->  ( <. X ,  Y >. O Z )  =  ( g  e.  ( Y H Z ) ,  f  e.  ( X H Y ) 
|->  ( g ( <. X ,  Y >.  .x. 
Z ) f ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   <.cop 3810   ` cfv 5447  (class class class)co 6074    e. cmpt2 6076   Basecbs 13462    Hom chom 13533  compcco 13534    Homf chomf 13884  compfccomf 13885
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 2417  ax-rep 4313  ax-sep 4323  ax-nul 4331  ax-pow 4370  ax-pr 4396  ax-un 4694
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2703  df-rex 2704  df-reu 2705  df-rab 2707  df-v 2951  df-sbc 3155  df-csb 3245  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-nul 3622  df-if 3733  df-pw 3794  df-sn 3813  df-pr 3814  df-op 3816  df-uni 4009  df-iun 4088  df-br 4206  df-opab 4260  df-mpt 4261  df-id 4491  df-xp 4877  df-rel 4878  df-cnv 4879  df-co 4880  df-dm 4881  df-rn 4882  df-res 4883  df-ima 4884  df-iota 5411  df-fun 5449  df-fn 5450  df-f 5451  df-f1 5452  df-fo 5453  df-f1o 5454  df-fv 5455  df-ov 6077  df-oprab 6078  df-mpt2 6079  df-1st 6342  df-2nd 6343  df-homf 13888  df-comf 13889
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