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Theorem comfffval 13883
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 5691 . . . . . . 7  |-  ( c  =  C  ->  ( Base `  c )  =  ( Base `  C
) )
3 comfffval.b . . . . . . 7  |-  B  =  ( Base `  C
)
42, 3syl6eqr 2458 . . . . . 6  |-  ( c  =  C  ->  ( Base `  c )  =  B )
54, 4xpeq12d 4866 . . . . 5  |-  ( c  =  C  ->  (
( Base `  c )  X.  ( Base `  c
) )  =  ( B  X.  B ) )
6 fveq2 5691 . . . . . . . 8  |-  ( c  =  C  ->  (  Hom  `  c )  =  (  Hom  `  C
) )
7 comfffval.h . . . . . . . 8  |-  H  =  (  Hom  `  C
)
86, 7syl6eqr 2458 . . . . . . 7  |-  ( c  =  C  ->  (  Hom  `  c )  =  H )
98oveqd 6061 . . . . . 6  |-  ( c  =  C  ->  (
( 2nd `  x
) (  Hom  `  c
) y )  =  ( ( 2nd `  x
) H y ) )
108fveq1d 5693 . . . . . 6  |-  ( c  =  C  ->  (
(  Hom  `  c ) `
 x )  =  ( H `  x
) )
11 fveq2 5691 . . . . . . . . 9  |-  ( c  =  C  ->  (comp `  c )  =  (comp `  C ) )
12 comfffval.x . . . . . . . . 9  |-  .x.  =  (comp `  C )
1311, 12syl6eqr 2458 . . . . . . . 8  |-  ( c  =  C  ->  (comp `  c )  =  .x.  )
1413oveqd 6061 . . . . . . 7  |-  ( c  =  C  ->  (
x (comp `  c
) y )  =  ( x  .x.  y
) )
1514oveqd 6061 . . . . . 6  |-  ( c  =  C  ->  (
g ( x (comp `  c ) y ) f )  =  ( g ( x  .x.  y ) f ) )
169, 10, 15mpt2eq123dv 6099 . . . . 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 6099 . . . 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 13855 . . . 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 5705 . . . . . . 7  |-  ( Base `  C )  e.  _V
203, 19eqeltri 2478 . . . . . 6  |-  B  e. 
_V
2120, 20xpex 4953 . . . . 5  |-  ( B  X.  B )  e. 
_V
2221, 20mpt2ex 6388 . . . 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 5769 . . 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 5685 . . . 4  |-  ( -.  C  e.  _V  ->  (compf `  C )  =  (/) )
25 fvprc 5685 . . . . . . . . 9  |-  ( -.  C  e.  _V  ->  (
Base `  C )  =  (/) )
263, 25syl5eq 2452 . . . . . . . 8  |-  ( -.  C  e.  _V  ->  B  =  (/) )
2726xpeq2d 4865 . . . . . . 7  |-  ( -.  C  e.  _V  ->  ( B  X.  B )  =  ( B  X.  (/) ) )
28 xp0 5254 . . . . . . 7  |-  ( B  X.  (/) )  =  (/)
2927, 28syl6eq 2456 . . . . . 6  |-  ( -.  C  e.  _V  ->  ( B  X.  B )  =  (/) )
30 mpt2eq12 6097 . . . . . 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 643 . . . . 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 6390 . . . . 5  |-  ( x  e.  (/) ,  y  e.  (/)  |->  ( g  e.  ( ( 2nd `  x
) H y ) ,  f  e.  ( H `  x ) 
|->  ( g ( x 
.x.  y ) f ) ) )  =  (/)
3331, 32syl6eq 2456 . . . 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 2443 . . 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 158 . 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 2428 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 1649    e. wcel 1721   _Vcvv 2920   (/)c0 3592    X. cxp 4839   ` cfv 5417  (class class class)co 6044    e. cmpt2 6046   2ndc2nd 6311   Basecbs 13428    Hom chom 13499  compcco 13500  compfccomf 13851
This theorem is referenced by:  comffval  13884  comfffval2  13886  comfffn  13889  comfeq  13891
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-reu 2677  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-1st 6312  df-2nd 6313  df-comf 13855
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