Theorem comfffval 16358

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 6191	. . . . . . 7 |- ( c = C -> ( Base ` c ) = ( Base ` C ) )
3		comfffval.b	. . . . . . 7 |- B = ( Base ` C )
4	2, 3	syl6eqr 2674	. . . . . 6 |- ( c = C -> ( Base ` c ) = B )
5	4	sqxpeqd 5141	. . . . 5 |- ( c = C -> ( ( Base ` c ) X. ( Base ` c ) ) = ( B X. B ) )
6		fveq2 6191	. . . . . . . 8 |- ( c = C -> ( Hom ` c ) = ( Hom ` C ) )
7		comfffval.h	. . . . . . . 8 |- H = ( Hom ` C )
8	6, 7	syl6eqr 2674	. . . . . . 7 |- ( c = C -> ( Hom ` c ) = H )
9	8	oveqd 6667	. . . . . 6 |- ( c = C -> ( ( 2nd ` x ) ( Hom ` c ) y ) = ( ( 2nd ` x ) H y ) )
10	8	fveq1d 6193	. . . . . 6 |- ( c = C -> ( ( Hom ` c ) ` x ) = ( H ` x ) )
11		fveq2 6191	. . . . . . . . 9 |- ( c = C -> (comp ` c ) = (comp ` C ) )
12		comfffval.x	. . . . . . . . 9 |- .x. = (comp ` C )
13	11, 12	syl6eqr 2674	. . . . . . . 8 |- ( c = C -> (comp ` c ) = .x. )
14	13	oveqd 6667	. . . . . . 7 |- ( c = C -> ( x (comp ` c ) y ) = ( x .x. y ) )
15	14	oveqd 6667	. . . . . 6 |- ( c = C -> ( g ( x (comp ` c ) y ) f ) = ( g ( x .x. y ) f ) )
16	9, 10, 15	mpt2eq123dv 6717	. . . . 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 6717	. . . 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 16332	. . . 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 6201	. . . . . . 7 |- ( Base ` C ) e. _V
20	3, 19	eqeltri 2697	. . . . . 6 |- B e. _V
21	20, 20	xpex 6962	. . . . 5 |- ( B X. B ) e. _V
22	21, 20	mpt2ex 7247	. . . 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 6282	. . 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 6185	. . . 4 |- ( -. C e. _V -> (compf ` C ) = (/) )
25		fvprc 6185	. . . . . . . . 9 |- ( -. C e. _V -> ( Base ` C ) = (/) )
26	3, 25	syl5eq 2668	. . . . . . . 8 |- ( -. C e. _V -> B = (/) )
27	26	xpeq2d 5139	. . . . . . 7 |- ( -. C e. _V -> ( B X. B ) = ( B X. (/) ) )
28		xp0 5552	. . . . . . 7 |- ( B X. (/) ) = (/)
29	27, 28	syl6eq 2672	. . . . . 6 |- ( -. C e. _V -> ( B X. B ) = (/) )
30		mpt2eq12 6715	. . . . . 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 693	. . . . 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 6725	. . . . 5 |- ( x e. (/) , y e. (/) |-> ( g e. ( ( 2nd ` x ) H y ) , f e. ( H ` x ) |-> ( g ( x .x. y ) f ) ) ) = (/)
33	31, 32	syl6eq 2672	. . . 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 2659	. . 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 176	. 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 2644	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 1483   e. wcel 1990   _Vcvv 3200   (/)c0 3915   X. cxp 5112   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   2ndc2nd 7167   Basecbs 15857   Hom chom 15952  compcco 15953  comp_fccomf 16328

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-comf 16332

This theorem is referenced by:  comffval  16359  comfffval2  16361  comfffn  16364  comfeq  16366