Theorem comfffval 15114

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 5872	. . . . . . 7 |- ( c = C -> ( Base ` c ) = ( Base ` C ) )
3		comfffval.b	. . . . . . 7 |- B = ( Base ` C )
4	2, 3	syl6eqr 2516	. . . . . 6 |- ( c = C -> ( Base ` c ) = B )
5	4	sqxpeqd 5034	. . . . 5 |- ( c = C -> ( ( Base ` c ) X. ( Base ` c ) ) = ( B X. B ) )
6		fveq2 5872	. . . . . . . 8 |- ( c = C -> ( Hom ` c ) = ( Hom ` C ) )
7		comfffval.h	. . . . . . . 8 |- H = ( Hom ` C )
8	6, 7	syl6eqr 2516	. . . . . . 7 |- ( c = C -> ( Hom ` c ) = H )
9	8	oveqd 6313	. . . . . 6 |- ( c = C -> ( ( 2nd ` x ) ( Hom ` c ) y ) = ( ( 2nd ` x ) H y ) )
10	8	fveq1d 5874	. . . . . 6 |- ( c = C -> ( ( Hom ` c ) ` x ) = ( H ` x ) )
11		fveq2 5872	. . . . . . . . 9 |- ( c = C -> (comp ` c ) = (comp ` C ) )
12		comfffval.x	. . . . . . . . 9 |- .x. = (comp ` C )
13	11, 12	syl6eqr 2516	. . . . . . . 8 |- ( c = C -> (comp ` c ) = .x. )
14	13	oveqd 6313	. . . . . . 7 |- ( c = C -> ( x (comp ` c ) y ) = ( x .x. y ) )
15	14	oveqd 6313	. . . . . 6 |- ( c = C -> ( g ( x (comp ` c ) y ) f ) = ( g ( x .x. y ) f ) )
16	9, 10, 15	mpt2eq123dv 6358	. . . . 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 6358	. . . 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 15088	. . . 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 5882	. . . . . . 7 |- ( Base ` C ) e. _V
20	3, 19	eqeltri 2541	. . . . . 6 |- B e. _V
21	20, 20	xpex 6603	. . . . 5 |- ( B X. B ) e. _V
22	21, 20	mpt2ex 6876	. . . 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 5956	. . 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 5866	. . . 4 |- ( -. C e. _V -> (compf ` C ) = (/) )
25		fvprc 5866	. . . . . . . . 9 |- ( -. C e. _V -> ( Base ` C ) = (/) )
26	3, 25	syl5eq 2510	. . . . . . . 8 |- ( -. C e. _V -> B = (/) )
27	26	xpeq2d 5032	. . . . . . 7 |- ( -. C e. _V -> ( B X. B ) = ( B X. (/) ) )
28		xp0 5432	. . . . . . 7 |- ( B X. (/) ) = (/)
29	27, 28	syl6eq 2514	. . . . . 6 |- ( -. C e. _V -> ( B X. B ) = (/) )
30		mpt2eq12 6356	. . . . . 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 6366	. . . . 5 |- ( x e. (/) , y e. (/) |-> ( g e. ( ( 2nd ` x ) H y ) , f e. ( H ` x ) |-> ( g ( x .x. y ) f ) ) ) = (/)
33	31, 32	syl6eq 2514	. . . 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 2501	. . 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 2486	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 1395   e. wcel 1819   _Vcvv 3109   (/)c0 3793   X. cxp 5006   ` cfv 5594  (class class class)co 6296   |-> cmpt2 6298   2ndc2nd 6798   Basecbs 14644   Hom chom 14723  compcco 14724  comp_fccomf 15084

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6799  df-2nd 6800  df-comf 15088

This theorem is referenced by:  comffval  15115  comfffval2  15117  comfffn  15120  comfeq  15122