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.

Step	Hyp	Ref	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 )
4	2, 3	syl6eqr 2493	. . . . . 6 |- ( c = C -> ( Base ` c ) = B )
5	4, 4	xpeq12d 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 )
8	6, 7	syl6eqr 2493	. . . . . . 7 |- ( c = C -> ( Hom ` c ) = H )
9	8	oveqd 6113	. . . . . 6 |- ( c = C -> ( ( 2nd ` x ) ( Hom ` c ) y ) = ( ( 2nd ` x ) H y ) )
10	8	fveq1d 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 )
13	11, 12	syl6eqr 2493	. . . . . . . 8 |- ( c = C -> (comp ` c ) = .x. )
14	13	oveqd 6113	. . . . . . 7 |- ( c = C -> ( x (comp ` c ) y ) = ( x .x. y ) )
15	14	oveqd 6113	. . . . . 6 |- ( c = C -> ( g ( x (comp ` c ) y ) f ) = ( g ( x .x. y ) f ) )
16	9, 10, 15	mpt2eq123dv 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 ) ) )
17	5, 4, 16	mpt2eq123dv 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
20	3, 19	eqeltri 2513	. . . . . 6 |- B e. _V
21	20, 20	xpex 6513	. . . . 5 |- ( B X. B ) e. _V
22	21, 20	mpt2ex 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
23	17, 18, 22	fvmpt 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 ) = (/) )
26	3, 25	syl5eq 2487	. . . . . . . 8 |- ( -. C e. _V -> B = (/) )
27	26	xpeq2d 4869	. . . . . . 7 |- ( -. C e. _V -> ( B X. B ) = ( B X. (/) ) )
28		xp0 5261	. . . . . . 7 |- ( B X. (/) ) = (/)
29	27, 28	syl6eq 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 ) ) ) )
31	29, 26, 30	syl2anc 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 ) ) ) = (/)
33	31, 32	syl6eq 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 ) ) ) = (/) )
34	24, 33	eqtr4d 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 ) ) ) )
35	23, 34	pm2.61i 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 ) ) )
36	1, 35	eqtri 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 ) ) )

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  comp_fccomf 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