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Theorem comfffval2 15316
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 2404 . . 3  |-  ( Hom  `  C )  =  ( Hom  `  C )
4 comfffval2.x . . 3  |-  .x.  =  (comp `  C )
51, 2, 3, 4comfffval 15313 . 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 6817 . . . . . 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 15307 . . . 4  |-  ( ( x  e.  ( B  X.  B )  /\  y  e.  B )  ->  ( ( 2nd `  x
) H y )  =  ( ( 2nd `  x ) ( Hom  `  C ) y ) )
11 xp1st 6816 . . . . . . . 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 15307 . . . . . 6  |-  ( ( x  e.  ( B  X.  B )  /\  y  e.  B )  ->  ( ( 1st `  x
) H ( 2nd `  x ) )  =  ( ( 1st `  x
) ( Hom  `  C
) ( 2nd `  x
) ) )
14 df-ov 6283 . . . . . 6  |-  ( ( 1st `  x ) H ( 2nd `  x
) )  =  ( H `  <. ( 1st `  x ) ,  ( 2nd `  x
) >. )
15 df-ov 6283 . . . . . 6  |-  ( ( 1st `  x ) ( Hom  `  C
) ( 2nd `  x
) )  =  ( ( Hom  `  C
) `  <. ( 1st `  x ) ,  ( 2nd `  x )
>. )
1613, 14, 153eqtr3g 2468 . . . . 5  |-  ( ( x  e.  ( B  X.  B )  /\  y  e.  B )  ->  ( H `  <. ( 1st `  x ) ,  ( 2nd `  x
) >. )  =  ( ( Hom  `  C
) `  <. ( 1st `  x ) ,  ( 2nd `  x )
>. ) )
17 1st2nd2 6823 . . . . . . 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 5855 . . . . 5  |-  ( ( x  e.  ( B  X.  B )  /\  y  e.  B )  ->  ( H `  x
)  =  ( H `
 <. ( 1st `  x
) ,  ( 2nd `  x ) >. )
)
2018fveq2d 5855 . . . . 5  |-  ( ( x  e.  ( B  X.  B )  /\  y  e.  B )  ->  ( ( Hom  `  C
) `  x )  =  ( ( Hom  `  C ) `  <. ( 1st `  x ) ,  ( 2nd `  x
) >. ) )
2116, 19, 203eqtr4d 2455 . . . 4  |-  ( ( x  e.  ( B  X.  B )  /\  y  e.  B )  ->  ( H `  x
)  =  ( ( Hom  `  C ) `  x ) )
22 eqidd 2405 . . . 4  |-  ( ( x  e.  ( B  X.  B )  /\  y  e.  B )  ->  ( g ( x 
.x.  y ) f )  =  ( g ( x  .x.  y
) f ) )
2310, 21, 22mpt2eq123dv 6342 . . 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 6345 . 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 2436 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 1407    e. wcel 1844   <.cop 3980    X. cxp 4823   ` cfv 5571  (class class class)co 6280    |-> cmpt2 6282   1stc1st 6784   2ndc2nd 6785   Basecbs 14843   Hom chom 14922  compcco 14923   Hom f chomf 15282  compfccomf 15283
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-rep 4509  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3063  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-op 3981  df-uni 4194  df-iun 4275  df-br 4398  df-opab 4456  df-mpt 4457  df-id 4740  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-f1 5576  df-fo 5577  df-f1o 5578  df-fv 5579  df-ov 6283  df-oprab 6284  df-mpt2 6285  df-1st 6786  df-2nd 6787  df-homf 15286  df-comf 15287
This theorem is referenced by: (None)
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