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Theorem comffval2 14974
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 2467 . . 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 14971 . 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 14964 . . 3  |-  ( ph  ->  ( Y H Z )  =  ( Y ( Hom  `  C
) Z ) )
119, 2, 3, 5, 6homfval 14964 . . 3  |-  ( ph  ->  ( X H Y )  =  ( X ( Hom  `  C
) Y ) )
12 eqidd 2468 . . 3  |-  ( ph  ->  ( g ( <. X ,  Y >.  .x. 
Z ) f )  =  ( g (
<. X ,  Y >.  .x. 
Z ) f ) )
1310, 11, 12mpt2eq123dv 6354 . 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 2511 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 1379    e. wcel 1767   <.cop 4039   ` cfv 5594  (class class class)co 6295    |-> cmpt2 6297   Basecbs 14506   Hom chom 14582  compcco 14583   Hom f chomf 14937  compfccomf 14938
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-1st 6795  df-2nd 6796  df-homf 14941  df-comf 14942
This theorem is referenced by: (None)
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