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Theorem comfffval2 14632
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 )
Assertion
Ref Expression
comfffval2  |-  O  =  ( x  e.  ( B  X.  B ) ,  y  e.  B  |->  ( g  e.  ( ( 2nd `  x
) H y ) ,  f  e.  ( H `  x ) 
|->  ( g ( x 
.x.  y ) f ) ) )
Distinct variable groups:    f, g, x, y, B    C, f,
g, x, y    .x. , f,
g, x
Allowed substitution hints:    .x. ( y)    H( x, y, f, g)    O( x, y, f, g)

Proof of Theorem comfffval2
StepHypRef Expression
1 comfffval2.o . . 3  |-  O  =  (compf `  C )
2 comfffval2.b . . 3  |-  B  =  ( Base `  C
)
3 eqid 2437 . . 3  |-  ( Hom  `  C )  =  ( Hom  `  C )
4 comfffval2.x . . 3  |-  .x.  =  (comp `  C )
51, 2, 3, 4comfffval 14629 . 2  |-  O  =  ( x  e.  ( B  X.  B ) ,  y  e.  B  |->  ( g  e.  ( ( 2nd `  x
) ( Hom  `  C
) y ) ,  f  e.  ( ( Hom  `  C ) `  x )  |->  ( g ( x  .x.  y
) f ) ) )
6 comfffval2.h . . . . 5  |-  H  =  ( Hom f  `  C )
7 xp2nd 6602 . . . . . 6  |-  ( x  e.  ( B  X.  B )  ->  ( 2nd `  x )  e.  B )
87adantr 465 . . . . 5  |-  ( ( x  e.  ( B  X.  B )  /\  y  e.  B )  ->  ( 2nd `  x
)  e.  B )
9 simpr 461 . . . . 5  |-  ( ( x  e.  ( B  X.  B )  /\  y  e.  B )  ->  y  e.  B )
106, 2, 3, 8, 9homfval 14623 . . . 4  |-  ( ( x  e.  ( B  X.  B )  /\  y  e.  B )  ->  ( ( 2nd `  x
) H y )  =  ( ( 2nd `  x ) ( Hom  `  C ) y ) )
11 xp1st 6601 . . . . . . . 8  |-  ( x  e.  ( B  X.  B )  ->  ( 1st `  x )  e.  B )
1211adantr 465 . . . . . . 7  |-  ( ( x  e.  ( B  X.  B )  /\  y  e.  B )  ->  ( 1st `  x
)  e.  B )
136, 2, 3, 12, 8homfval 14623 . . . . . 6  |-  ( ( x  e.  ( B  X.  B )  /\  y  e.  B )  ->  ( ( 1st `  x
) H ( 2nd `  x ) )  =  ( ( 1st `  x
) ( Hom  `  C
) ( 2nd `  x
) ) )
14 df-ov 6089 . . . . . 6  |-  ( ( 1st `  x ) H ( 2nd `  x
) )  =  ( H `  <. ( 1st `  x ) ,  ( 2nd `  x
) >. )
15 df-ov 6089 . . . . . 6  |-  ( ( 1st `  x ) ( Hom  `  C
) ( 2nd `  x
) )  =  ( ( Hom  `  C
) `  <. ( 1st `  x ) ,  ( 2nd `  x )
>. )
1613, 14, 153eqtr3g 2492 . . . . 5  |-  ( ( x  e.  ( B  X.  B )  /\  y  e.  B )  ->  ( H `  <. ( 1st `  x ) ,  ( 2nd `  x
) >. )  =  ( ( Hom  `  C
) `  <. ( 1st `  x ) ,  ( 2nd `  x )
>. ) )
17 1st2nd2 6608 . . . . . . 7  |-  ( x  e.  ( B  X.  B )  ->  x  =  <. ( 1st `  x
) ,  ( 2nd `  x ) >. )
1817adantr 465 . . . . . 6  |-  ( ( x  e.  ( B  X.  B )  /\  y  e.  B )  ->  x  =  <. ( 1st `  x ) ,  ( 2nd `  x
) >. )
1918fveq2d 5688 . . . . 5  |-  ( ( x  e.  ( B  X.  B )  /\  y  e.  B )  ->  ( H `  x
)  =  ( H `
 <. ( 1st `  x
) ,  ( 2nd `  x ) >. )
)
2018fveq2d 5688 . . . . 5  |-  ( ( x  e.  ( B  X.  B )  /\  y  e.  B )  ->  ( ( Hom  `  C
) `  x )  =  ( ( Hom  `  C ) `  <. ( 1st `  x ) ,  ( 2nd `  x
) >. ) )
2116, 19, 203eqtr4d 2479 . . . 4  |-  ( ( x  e.  ( B  X.  B )  /\  y  e.  B )  ->  ( H `  x
)  =  ( ( Hom  `  C ) `  x ) )
22 eqidd 2438 . . . 4  |-  ( ( x  e.  ( B  X.  B )  /\  y  e.  B )  ->  ( g ( x 
.x.  y ) f )  =  ( g ( x  .x.  y
) f ) )
2310, 21, 22mpt2eq123dv 6143 . . 3  |-  ( ( x  e.  ( B  X.  B )  /\  y  e.  B )  ->  ( g  e.  ( ( 2nd `  x
) H y ) ,  f  e.  ( H `  x ) 
|->  ( g ( x 
.x.  y ) f ) )  =  ( g  e.  ( ( 2nd `  x ) ( Hom  `  C
) y ) ,  f  e.  ( ( Hom  `  C ) `  x )  |->  ( g ( x  .x.  y
) f ) ) )
2423mpt2eq3ia 6146 . 2  |-  ( x  e.  ( B  X.  B ) ,  y  e.  B  |->  ( g  e.  ( ( 2nd `  x ) H y ) ,  f  e.  ( H `  x
)  |->  ( g ( x  .x.  y ) f ) ) )  =  ( x  e.  ( B  X.  B
) ,  y  e.  B  |->  ( g  e.  ( ( 2nd `  x
) ( Hom  `  C
) y ) ,  f  e.  ( ( Hom  `  C ) `  x )  |->  ( g ( x  .x.  y
) f ) ) )
255, 24eqtr4i 2460 1  |-  O  =  ( x  e.  ( B  X.  B ) ,  y  e.  B  |->  ( g  e.  ( ( 2nd `  x
) H y ) ,  f  e.  ( H `  x ) 
|->  ( g ( x 
.x.  y ) f ) ) )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    = wceq 1369    e. wcel 1756   <.cop 3876    X. cxp 4830   ` cfv 5411  (class class class)co 6086    e. cmpt2 6088   1stc1st 6570   2ndc2nd 6571   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 2418  ax-rep 4396  ax-sep 4406  ax-nul 4414  ax-pow 4463  ax-pr 4524  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 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2714  df-rex 2715  df-reu 2716  df-rab 2718  df-v 2968  df-sbc 3180  df-csb 3282  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3631  df-if 3785  df-pw 3855  df-sn 3871  df-pr 3873  df-op 3877  df-uni 4085  df-iun 4166  df-br 4286  df-opab 4344  df-mpt 4345  df-id 4628  df-xp 4838  df-rel 4839  df-cnv 4840  df-co 4841  df-dm 4842  df-rn 4843  df-res 4844  df-ima 4845  df-iota 5374  df-fun 5413  df-fn 5414  df-f 5415  df-f1 5416  df-fo 5417  df-f1o 5418  df-fv 5419  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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