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Theorem comfffval 13700
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 5608 . . . . . . 7  |-  ( c  =  C  ->  ( Base `  c )  =  ( Base `  C
) )
3 comfffval.b . . . . . . 7  |-  B  =  ( Base `  C
)
42, 3syl6eqr 2408 . . . . . 6  |-  ( c  =  C  ->  ( Base `  c )  =  B )
54, 4xpeq12d 4796 . . . . 5  |-  ( c  =  C  ->  (
( Base `  c )  X.  ( Base `  c
) )  =  ( B  X.  B ) )
6 fveq2 5608 . . . . . . . 8  |-  ( c  =  C  ->  (  Hom  `  c )  =  (  Hom  `  C
) )
7 comfffval.h . . . . . . . 8  |-  H  =  (  Hom  `  C
)
86, 7syl6eqr 2408 . . . . . . 7  |-  ( c  =  C  ->  (  Hom  `  c )  =  H )
98oveqd 5962 . . . . . 6  |-  ( c  =  C  ->  (
( 2nd `  x
) (  Hom  `  c
) y )  =  ( ( 2nd `  x
) H y ) )
108fveq1d 5610 . . . . . 6  |-  ( c  =  C  ->  (
(  Hom  `  c ) `
 x )  =  ( H `  x
) )
11 fveq2 5608 . . . . . . . . 9  |-  ( c  =  C  ->  (comp `  c )  =  (comp `  C ) )
12 comfffval.x . . . . . . . . 9  |-  .x.  =  (comp `  C )
1311, 12syl6eqr 2408 . . . . . . . 8  |-  ( c  =  C  ->  (comp `  c )  =  .x.  )
1413oveqd 5962 . . . . . . 7  |-  ( c  =  C  ->  (
x (comp `  c
) y )  =  ( x  .x.  y
) )
1514oveqd 5962 . . . . . 6  |-  ( c  =  C  ->  (
g ( x (comp `  c ) y ) f )  =  ( g ( x  .x.  y ) f ) )
169, 10, 15mpt2eq123dv 5997 . . . . 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 5997 . . . 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 13672 . . . 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 5622 . . . . . . 7  |-  ( Base `  C )  e.  _V
203, 19eqeltri 2428 . . . . . 6  |-  B  e. 
_V
2120, 20xpex 4883 . . . . 5  |-  ( B  X.  B )  e. 
_V
2221, 20mpt2ex 6285 . . . 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 5685 . . 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 5602 . . . 4  |-  ( -.  C  e.  _V  ->  (compf `  C )  =  (/) )
25 fvprc 5602 . . . . . . . . 9  |-  ( -.  C  e.  _V  ->  (
Base `  C )  =  (/) )
263, 25syl5eq 2402 . . . . . . . 8  |-  ( -.  C  e.  _V  ->  B  =  (/) )
2726xpeq2d 4795 . . . . . . 7  |-  ( -.  C  e.  _V  ->  ( B  X.  B )  =  ( B  X.  (/) ) )
28 xp0 5180 . . . . . . 7  |-  ( B  X.  (/) )  =  (/)
2927, 28syl6eq 2406 . . . . . 6  |-  ( -.  C  e.  _V  ->  ( B  X.  B )  =  (/) )
30 mpt2eq12 5995 . . . . . 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 642 . . . . 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 6286 . . . . 5  |-  ( x  e.  (/) ,  y  e.  (/)  |->  ( g  e.  ( ( 2nd `  x
) H y ) ,  f  e.  ( H `  x ) 
|->  ( g ( x 
.x.  y ) f ) ) )  =  (/)
3331, 32syl6eq 2406 . . . 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 2393 . . 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 156 . 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 2378 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:   -. wn 3    = wceq 1642    e. wcel 1710   _Vcvv 2864   (/)c0 3531    X. cxp 4769   ` cfv 5337  (class class class)co 5945    e. cmpt2 5947   2ndc2nd 6208   Basecbs 13245    Hom chom 13316  compcco 13317  compfccomf 13668
This theorem is referenced by:  comffval  13701  comfffval2  13703  comfffn  13706  comfeq  13708
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-id 4391  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-1st 6209  df-2nd 6210  df-comf 13672
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