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Theorem comfffval2 13620
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 2296 . . 3  |-  (  Hom  `  C )  =  (  Hom  `  C )
4 comfffval2.x . . 3  |-  .x.  =  (comp `  C )
51, 2, 3, 4comfffval 13617 . 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 6166 . . . . . 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 13611 . . . 4  |-  ( ( x  e.  ( B  X.  B )  /\  y  e.  B )  ->  ( ( 2nd `  x
) H y )  =  ( ( 2nd `  x ) (  Hom  `  C ) y ) )
11 xp1st 6165 . . . . . . . 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 13611 . . . . . 6  |-  ( ( x  e.  ( B  X.  B )  /\  y  e.  B )  ->  ( ( 1st `  x
) H ( 2nd `  x ) )  =  ( ( 1st `  x
) (  Hom  `  C
) ( 2nd `  x
) ) )
14 df-ov 5877 . . . . . 6  |-  ( ( 1st `  x ) H ( 2nd `  x
) )  =  ( H `  <. ( 1st `  x ) ,  ( 2nd `  x
) >. )
15 df-ov 5877 . . . . . 6  |-  ( ( 1st `  x ) (  Hom  `  C
) ( 2nd `  x
) )  =  ( (  Hom  `  C
) `  <. ( 1st `  x ) ,  ( 2nd `  x )
>. )
1613, 14, 153eqtr3g 2351 . . . . 5  |-  ( ( x  e.  ( B  X.  B )  /\  y  e.  B )  ->  ( H `  <. ( 1st `  x ) ,  ( 2nd `  x
) >. )  =  ( (  Hom  `  C
) `  <. ( 1st `  x ) ,  ( 2nd `  x )
>. ) )
17 1st2nd2 6175 . . . . . . 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 5545 . . . . 5  |-  ( ( x  e.  ( B  X.  B )  /\  y  e.  B )  ->  ( H `  x
)  =  ( H `
 <. ( 1st `  x
) ,  ( 2nd `  x ) >. )
)
2018fveq2d 5545 . . . . 5  |-  ( ( x  e.  ( B  X.  B )  /\  y  e.  B )  ->  ( (  Hom  `  C
) `  x )  =  ( (  Hom  `  C ) `  <. ( 1st `  x ) ,  ( 2nd `  x
) >. ) )
2116, 19, 203eqtr4d 2338 . . . 4  |-  ( ( x  e.  ( B  X.  B )  /\  y  e.  B )  ->  ( H `  x
)  =  ( (  Hom  `  C ) `  x ) )
22 eqidd 2297 . . . 4  |-  ( ( x  e.  ( B  X.  B )  /\  y  e.  B )  ->  ( g ( x 
.x.  y ) f )  =  ( g ( x  .x.  y
) f ) )
2310, 21, 22mpt2eq123dv 5926 . . 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 5929 . 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 2319 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 1632    e. wcel 1696   <.cop 3656    X. cxp 4703   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876   1stc1st 6136   2ndc2nd 6137   Basecbs 13164    Hom chom 13235  compcco 13236    Homf chomf 13584  compfccomf 13585
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-homf 13588  df-comf 13589
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