MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  comfffval2 Unicode version

Theorem comfffval2 13604
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 )
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 2283 . . 3  |-  (  Hom  `  C )  =  (  Hom  `  C )
4 comfffval2.x . . 3  |-  .x.  =  (comp `  C )
51, 2, 3, 4comfffval 13601 . 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  =  (  Homf 
`  C )
7 xp2nd 6150 . . . . . 6  |-  ( x  e.  ( B  X.  B )  ->  ( 2nd `  x )  e.  B )
87adantr 451 . . . . 5  |-  ( ( x  e.  ( B  X.  B )  /\  y  e.  B )  ->  ( 2nd `  x
)  e.  B )
9 simpr 447 . . . . 5  |-  ( ( x  e.  ( B  X.  B )  /\  y  e.  B )  ->  y  e.  B )
106, 2, 3, 8, 9homfval 13595 . . . 4  |-  ( ( x  e.  ( B  X.  B )  /\  y  e.  B )  ->  ( ( 2nd `  x
) H y )  =  ( ( 2nd `  x ) (  Hom  `  C ) y ) )
11 xp1st 6149 . . . . . . . 8  |-  ( x  e.  ( B  X.  B )  ->  ( 1st `  x )  e.  B )
1211adantr 451 . . . . . . 7  |-  ( ( x  e.  ( B  X.  B )  /\  y  e.  B )  ->  ( 1st `  x
)  e.  B )
136, 2, 3, 12, 8homfval 13595 . . . . . 6  |-  ( ( x  e.  ( B  X.  B )  /\  y  e.  B )  ->  ( ( 1st `  x
) H ( 2nd `  x ) )  =  ( ( 1st `  x
) (  Hom  `  C
) ( 2nd `  x
) ) )
14 df-ov 5861 . . . . . 6  |-  ( ( 1st `  x ) H ( 2nd `  x
) )  =  ( H `  <. ( 1st `  x ) ,  ( 2nd `  x
) >. )
15 df-ov 5861 . . . . . 6  |-  ( ( 1st `  x ) (  Hom  `  C
) ( 2nd `  x
) )  =  ( (  Hom  `  C
) `  <. ( 1st `  x ) ,  ( 2nd `  x )
>. )
1613, 14, 153eqtr3g 2338 . . . . 5  |-  ( ( x  e.  ( B  X.  B )  /\  y  e.  B )  ->  ( H `  <. ( 1st `  x ) ,  ( 2nd `  x
) >. )  =  ( (  Hom  `  C
) `  <. ( 1st `  x ) ,  ( 2nd `  x )
>. ) )
17 1st2nd2 6159 . . . . . . 7  |-  ( x  e.  ( B  X.  B )  ->  x  =  <. ( 1st `  x
) ,  ( 2nd `  x ) >. )
1817adantr 451 . . . . . 6  |-  ( ( x  e.  ( B  X.  B )  /\  y  e.  B )  ->  x  =  <. ( 1st `  x ) ,  ( 2nd `  x
) >. )
1918fveq2d 5529 . . . . 5  |-  ( ( x  e.  ( B  X.  B )  /\  y  e.  B )  ->  ( H `  x
)  =  ( H `
 <. ( 1st `  x
) ,  ( 2nd `  x ) >. )
)
2018fveq2d 5529 . . . . 5  |-  ( ( x  e.  ( B  X.  B )  /\  y  e.  B )  ->  ( (  Hom  `  C
) `  x )  =  ( (  Hom  `  C ) `  <. ( 1st `  x ) ,  ( 2nd `  x
) >. ) )
2116, 19, 203eqtr4d 2325 . . . 4  |-  ( ( x  e.  ( B  X.  B )  /\  y  e.  B )  ->  ( H `  x
)  =  ( (  Hom  `  C ) `  x ) )
22 eqidd 2284 . . . 4  |-  ( ( x  e.  ( B  X.  B )  /\  y  e.  B )  ->  ( g ( x 
.x.  y ) f )  =  ( g ( x  .x.  y
) f ) )
2310, 21, 22mpt2eq123dv 5910 . . 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 5913 . 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 2306 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 set class
Syntax hints:    /\ wa 358    = wceq 1623    e. wcel 1684   <.cop 3643    X. cxp 4687   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   1stc1st 6120   2ndc2nd 6121   Basecbs 13148    Hom chom 13219  compcco 13220    Homf chomf 13568  compfccomf 13569
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-homf 13572  df-comf 13573
  Copyright terms: Public domain W3C validator