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Theorem comffval2 13855
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 2387 . . 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 13852 . 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 13845 . . 3  |-  ( ph  ->  ( Y H Z )  =  ( Y (  Hom  `  C
) Z ) )
119, 2, 3, 5, 6homfval 13845 . . 3  |-  ( ph  ->  ( X H Y )  =  ( X (  Hom  `  C
) Y ) )
12 eqidd 2388 . . 3  |-  ( ph  ->  ( g ( <. X ,  Y >.  .x. 
Z ) f )  =  ( g (
<. X ,  Y >.  .x. 
Z ) f ) )
1310, 11, 12mpt2eq123dv 6075 . 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 2422 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 1649    e. wcel 1717   <.cop 3760   ` cfv 5394  (class class class)co 6020    e. cmpt2 6022   Basecbs 13396    Hom chom 13467  compcco 13468    Homf chomf 13818  compfccomf 13819
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-reu 2656  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-homf 13822  df-comf 13823
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