Theorem comfffval 15615

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 5870	. . . . . . 7 |- ( c = C -> ( Base ` c ) = ( Base ` C ) )
3		comfffval.b	. . . . . . 7 |- B = ( Base ` C )
4	2, 3	syl6eqr 2505	. . . . . 6 |- ( c = C -> ( Base ` c ) = B )
5	4	sqxpeqd 4863	. . . . 5 |- ( c = C -> ( ( Base ` c ) X. ( Base ` c ) ) = ( B X. B ) )
6		fveq2 5870	. . . . . . . 8 |- ( c = C -> ( Hom ` c ) = ( Hom ` C ) )
7		comfffval.h	. . . . . . . 8 |- H = ( Hom ` C )
8	6, 7	syl6eqr 2505	. . . . . . 7 |- ( c = C -> ( Hom ` c ) = H )
9	8	oveqd 6312	. . . . . 6 |- ( c = C -> ( ( 2nd ` x ) ( Hom ` c ) y ) = ( ( 2nd ` x ) H y ) )
10	8	fveq1d 5872	. . . . . 6 |- ( c = C -> ( ( Hom ` c ) ` x ) = ( H ` x ) )
11		fveq2 5870	. . . . . . . . 9 |- ( c = C -> (comp ` c ) = (comp ` C ) )
12		comfffval.x	. . . . . . . . 9 |- .x. = (comp ` C )
13	11, 12	syl6eqr 2505	. . . . . . . 8 |- ( c = C -> (comp ` c ) = .x. )
14	13	oveqd 6312	. . . . . . 7 |- ( c = C -> ( x (comp ` c ) y ) = ( x .x. y ) )
15	14	oveqd 6312	. . . . . 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 15589	. . . 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 5880	. . . . . . 7 |- ( Base ` C ) e. _V
20	3, 19	eqeltri 2527	. . . . . 6 |- B e. _V
21	20, 20	xpex 6600	. . . . 5 |- ( B X. B ) e. _V
22	21, 20	mpt2ex 6875	. . . 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 5953	. . 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 5864	. . . 4 |- ( -. C e. _V -> (compf ` C ) = (/) )
25		fvprc 5864	. . . . . . . . 9 |- ( -. C e. _V -> ( Base ` C ) = (/) )
26	3, 25	syl5eq 2499	. . . . . . . 8 |- ( -. C e. _V -> B = (/) )
27	26	xpeq2d 4861	. . . . . . 7 |- ( -. C e. _V -> ( B X. B ) = ( B X. (/) ) )
28		xp0 5258	. . . . . . 7 |- ( B X. (/) ) = (/)
29	27, 28	syl6eq 2503	. . . . . 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 667	. . . . 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 2503	. . . 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 2490	. . 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 168	. 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 2475	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 1446   e. wcel 1889   _Vcvv 3047   (/)c0 3733   X. cxp 4835   ` cfv 5585  (class class class)co 6295   |-> cmpt2 6297   2ndc2nd 6797   Basecbs 15133   Hom chom 15213  compcco 15214  comp_fccomf 15585

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-rep 4518  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-ral 2744  df-rex 2745  df-reu 2746  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-op 3977  df-uni 4202  df-iun 4283  df-br 4406  df-opab 4465  df-mpt 4466  df-id 4752  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-1st 6798  df-2nd 6799  df-comf 15589

This theorem is referenced by:  comffval  15616  comfffval2  15618  comfffn  15621  comfeq  15623