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Theorem comfffval 14642
Description: Value of the functionalized composition operation. (Contributed by Mario Carneiro, 4-Jan-2017.)
Hypotheses
Ref Expression
comfffval.o  |-  O  =  (compf `  C )
comfffval.b  |-  B  =  ( Base `  C
)
comfffval.h  |-  H  =  ( Hom  `  C
)
comfffval.x  |-  .x.  =  (comp `  C )
Assertion
Ref Expression
comfffval  |-  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:    x, y, B    f, g, x, y, C    .x. , f, g, x   
f, H, g, x
Allowed substitution hints:    B( f, g)    .x. ( y)    H( y)    O( x, y, f, g)

Proof of Theorem comfffval
Dummy variable  c is distinct from all other variables.
StepHypRef Expression
1 comfffval.o . 2  |-  O  =  (compf `  C )
2 fveq2 5696 . . . . . . 7  |-  ( c  =  C  ->  ( Base `  c )  =  ( Base `  C
) )
3 comfffval.b . . . . . . 7  |-  B  =  ( Base `  C
)
42, 3syl6eqr 2493 . . . . . 6  |-  ( c  =  C  ->  ( Base `  c )  =  B )
54, 4xpeq12d 4870 . . . . 5  |-  ( c  =  C  ->  (
( Base `  c )  X.  ( Base `  c
) )  =  ( B  X.  B ) )
6 fveq2 5696 . . . . . . . 8  |-  ( c  =  C  ->  ( Hom  `  c )  =  ( Hom  `  C
) )
7 comfffval.h . . . . . . . 8  |-  H  =  ( Hom  `  C
)
86, 7syl6eqr 2493 . . . . . . 7  |-  ( c  =  C  ->  ( Hom  `  c )  =  H )
98oveqd 6113 . . . . . 6  |-  ( c  =  C  ->  (
( 2nd `  x
) ( Hom  `  c
) y )  =  ( ( 2nd `  x
) H y ) )
108fveq1d 5698 . . . . . 6  |-  ( c  =  C  ->  (
( Hom  `  c ) `
 x )  =  ( H `  x
) )
11 fveq2 5696 . . . . . . . . 9  |-  ( c  =  C  ->  (comp `  c )  =  (comp `  C ) )
12 comfffval.x . . . . . . . . 9  |-  .x.  =  (comp `  C )
1311, 12syl6eqr 2493 . . . . . . . 8  |-  ( c  =  C  ->  (comp `  c )  =  .x.  )
1413oveqd 6113 . . . . . . 7  |-  ( c  =  C  ->  (
x (comp `  c
) y )  =  ( x  .x.  y
) )
1514oveqd 6113 . . . . . 6  |-  ( c  =  C  ->  (
g ( x (comp `  c ) y ) f )  =  ( g ( x  .x.  y ) f ) )
169, 10, 15mpt2eq123dv 6153 . . . . 5  |-  ( c  =  C  ->  (
g  e.  ( ( 2nd `  x ) ( Hom  `  c
) y ) ,  f  e.  ( ( Hom  `  c ) `  x )  |->  ( g ( x (comp `  c ) y ) f ) )  =  ( g  e.  ( ( 2nd `  x
) H y ) ,  f  e.  ( H `  x ) 
|->  ( g ( x 
.x.  y ) f ) ) )
175, 4, 16mpt2eq123dv 6153 . . . 4  |-  ( c  =  C  ->  (
x  e.  ( (
Base `  c )  X.  ( Base `  c
) ) ,  y  e.  ( Base `  c
)  |->  ( g  e.  ( ( 2nd `  x
) ( Hom  `  c
) y ) ,  f  e.  ( ( Hom  `  c ) `  x )  |->  ( g ( x (comp `  c ) y ) f ) ) )  =  ( x  e.  ( B  X.  B
) ,  y  e.  B  |->  ( g  e.  ( ( 2nd `  x
) H y ) ,  f  e.  ( H `  x ) 
|->  ( g ( x 
.x.  y ) f ) ) ) )
18 df-comf 14614 . . . 4  |- compf  =  ( c  e. 
_V  |->  ( x  e.  ( ( Base `  c
)  X.  ( Base `  c ) ) ,  y  e.  ( Base `  c )  |->  ( g  e.  ( ( 2nd `  x ) ( Hom  `  c ) y ) ,  f  e.  ( ( Hom  `  c
) `  x )  |->  ( g ( x (comp `  c )
y ) f ) ) ) )
19 fvex 5706 . . . . . . 7  |-  ( Base `  C )  e.  _V
203, 19eqeltri 2513 . . . . . 6  |-  B  e. 
_V
2120, 20xpex 6513 . . . . 5  |-  ( B  X.  B )  e. 
_V
2221, 20mpt2ex 6655 . . . 4  |-  ( x  e.  ( B  X.  B ) ,  y  e.  B  |->  ( g  e.  ( ( 2nd `  x ) H y ) ,  f  e.  ( H `  x
)  |->  ( g ( x  .x.  y ) f ) ) )  e.  _V
2317, 18, 22fvmpt 5779 . . 3  |-  ( C  e.  _V  ->  (compf `  C
)  =  ( x  e.  ( B  X.  B ) ,  y  e.  B  |->  ( g  e.  ( ( 2nd `  x ) H y ) ,  f  e.  ( H `  x
)  |->  ( g ( x  .x.  y ) f ) ) ) )
24 fvprc 5690 . . . 4  |-  ( -.  C  e.  _V  ->  (compf `  C )  =  (/) )
25 fvprc 5690 . . . . . . . . 9  |-  ( -.  C  e.  _V  ->  (
Base `  C )  =  (/) )
263, 25syl5eq 2487 . . . . . . . 8  |-  ( -.  C  e.  _V  ->  B  =  (/) )
2726xpeq2d 4869 . . . . . . 7  |-  ( -.  C  e.  _V  ->  ( B  X.  B )  =  ( B  X.  (/) ) )
28 xp0 5261 . . . . . . 7  |-  ( B  X.  (/) )  =  (/)
2927, 28syl6eq 2491 . . . . . 6  |-  ( -.  C  e.  _V  ->  ( B  X.  B )  =  (/) )
30 mpt2eq12 6151 . . . . . 6  |-  ( ( ( B  X.  B
)  =  (/)  /\  B  =  (/) )  ->  (
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.  (/) ,  y  e.  (/)  |->  ( g  e.  ( ( 2nd `  x
) H y ) ,  f  e.  ( H `  x ) 
|->  ( g ( x 
.x.  y ) f ) ) ) )
3129, 26, 30syl2anc 661 . . . . 5  |-  ( -.  C  e.  _V  ->  ( 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.  (/) ,  y  e.  (/)  |->  ( g  e.  ( ( 2nd `  x
) H y ) ,  f  e.  ( H `  x ) 
|->  ( g ( x 
.x.  y ) f ) ) ) )
32 mpt20 6161 . . . . 5  |-  ( x  e.  (/) ,  y  e.  (/)  |->  ( g  e.  ( ( 2nd `  x
) H y ) ,  f  e.  ( H `  x ) 
|->  ( g ( x 
.x.  y ) f ) ) )  =  (/)
3331, 32syl6eq 2491 . . . 4  |-  ( -.  C  e.  _V  ->  ( x  e.  ( B  X.  B ) ,  y  e.  B  |->  ( g  e.  ( ( 2nd `  x ) H y ) ,  f  e.  ( H `
 x )  |->  ( g ( x  .x.  y ) f ) ) )  =  (/) )
3424, 33eqtr4d 2478 . . 3  |-  ( -.  C  e.  _V  ->  (compf `  C )  =  ( x  e.  ( B  X.  B ) ,  y  e.  B  |->  ( g  e.  ( ( 2nd `  x ) H y ) ,  f  e.  ( H `
 x )  |->  ( g ( x  .x.  y ) f ) ) ) )
3523, 34pm2.61i 164 . 2  |-  (compf `  C
)  =  ( x  e.  ( B  X.  B ) ,  y  e.  B  |->  ( g  e.  ( ( 2nd `  x ) H y ) ,  f  e.  ( H `  x
)  |->  ( g ( x  .x.  y ) f ) ) )
361, 35eqtri 2463 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:   -. wn 3    = wceq 1369    e. wcel 1756   _Vcvv 2977   (/)c0 3642    X. cxp 4843   ` cfv 5423  (class class class)co 6096    e. cmpt2 6098   2ndc2nd 6581   Basecbs 14179   Hom chom 14254  compcco 14255  compfccomf 14610
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 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377
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 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-reu 2727  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-1st 6582  df-2nd 6583  df-comf 14614
This theorem is referenced by:  comffval  14643  comfffval2  14645  comfffn  14648  comfeq  14650
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