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Theorem comfffval2 14948
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 2462 . . 3  |-  ( Hom  `  C )  =  ( Hom  `  C )
4 comfffval2.x . . 3  |-  .x.  =  (comp `  C )
51, 2, 3, 4comfffval 14945 . 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 6807 . . . . . 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 14939 . . . 4  |-  ( ( x  e.  ( B  X.  B )  /\  y  e.  B )  ->  ( ( 2nd `  x
) H y )  =  ( ( 2nd `  x ) ( Hom  `  C ) y ) )
11 xp1st 6806 . . . . . . . 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 14939 . . . . . 6  |-  ( ( x  e.  ( B  X.  B )  /\  y  e.  B )  ->  ( ( 1st `  x
) H ( 2nd `  x ) )  =  ( ( 1st `  x
) ( Hom  `  C
) ( 2nd `  x
) ) )
14 df-ov 6280 . . . . . 6  |-  ( ( 1st `  x ) H ( 2nd `  x
) )  =  ( H `  <. ( 1st `  x ) ,  ( 2nd `  x
) >. )
15 df-ov 6280 . . . . . 6  |-  ( ( 1st `  x ) ( Hom  `  C
) ( 2nd `  x
) )  =  ( ( Hom  `  C
) `  <. ( 1st `  x ) ,  ( 2nd `  x )
>. )
1613, 14, 153eqtr3g 2526 . . . . 5  |-  ( ( x  e.  ( B  X.  B )  /\  y  e.  B )  ->  ( H `  <. ( 1st `  x ) ,  ( 2nd `  x
) >. )  =  ( ( Hom  `  C
) `  <. ( 1st `  x ) ,  ( 2nd `  x )
>. ) )
17 1st2nd2 6813 . . . . . . 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 5863 . . . . 5  |-  ( ( x  e.  ( B  X.  B )  /\  y  e.  B )  ->  ( H `  x
)  =  ( H `
 <. ( 1st `  x
) ,  ( 2nd `  x ) >. )
)
2018fveq2d 5863 . . . . 5  |-  ( ( x  e.  ( B  X.  B )  /\  y  e.  B )  ->  ( ( Hom  `  C
) `  x )  =  ( ( Hom  `  C ) `  <. ( 1st `  x ) ,  ( 2nd `  x
) >. ) )
2116, 19, 203eqtr4d 2513 . . . 4  |-  ( ( x  e.  ( B  X.  B )  /\  y  e.  B )  ->  ( H `  x
)  =  ( ( Hom  `  C ) `  x ) )
22 eqidd 2463 . . . 4  |-  ( ( x  e.  ( B  X.  B )  /\  y  e.  B )  ->  ( g ( x 
.x.  y ) f )  =  ( g ( x  .x.  y
) f ) )
2310, 21, 22mpt2eq123dv 6336 . . 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 6339 . 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 2494 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 1374    e. wcel 1762   <.cop 4028    X. cxp 4992   ` cfv 5581  (class class class)co 6277    |-> cmpt2 6279   1stc1st 6774   2ndc2nd 6775   Basecbs 14481   Hom chom 14557  compcco 14558   Hom f chomf 14912  compfccomf 14913
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-reu 2816  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-id 4790  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-1st 6776  df-2nd 6777  df-homf 14916  df-comf 14917
This theorem is referenced by: (None)
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