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Theorem comffval2 14633
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  =  ( Hom f  `  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 2438 . . 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 14630 . 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  =  ( Hom f  `  C )
109, 2, 3, 6, 7homfval 14623 . . 3  |-  ( ph  ->  ( Y H Z )  =  ( Y ( Hom  `  C
) Z ) )
119, 2, 3, 5, 6homfval 14623 . . 3  |-  ( ph  ->  ( X H Y )  =  ( X ( Hom  `  C
) Y ) )
12 eqidd 2439 . . 3  |-  ( ph  ->  ( g ( <. X ,  Y >.  .x. 
Z ) f )  =  ( g (
<. X ,  Y >.  .x. 
Z ) f ) )
1310, 11, 12mpt2eq123dv 6143 . 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 2473 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 setvar class
Syntax hints:    -> wi 4    = wceq 1369    e. wcel 1756   <.cop 3878   ` cfv 5413  (class class class)co 6086    e. cmpt2 6088   Basecbs 14166   Hom chom 14241  compcco 14242   Hom f chomf 14596  compfccomf 14597
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-reu 2717  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-1st 6572  df-2nd 6573  df-homf 14600  df-comf 14601
This theorem is referenced by: (None)
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