Theorem comfffval2 16361

Description: Value of the functionalized composition operation. (Contributed by Mario Carneiro, 4-Jan-2017.)

Hypotheses

Ref	Expression
comfffval2.o	|- O = (compf ` C )
comfffval2.b	|- B = ( Base ` C )
comfffval2.h	|- H = ( Hom f ` C )
comfffval2.x	|- .x. = (comp ` C )

Assertion

Ref	Expression
comfffval2	|- 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:   f, g, x, y, B   C, f, g, x, y   .x. , f, g, x

Allowed substitution hints:   .x. ( y)   H( x, y, f, g)   O( x, y, f, g)

Proof of Theorem comfffval2

Step	Hyp	Ref	Expression
1		comfffval2.o	. . 3 |- O = (compf ` C )
2		comfffval2.b	. . 3 |- B = ( Base ` C )
3		eqid 2622	. . 3 |- ( Hom ` C ) = ( Hom ` C )
4		comfffval2.x	. . 3 |- .x. = (comp ` C )
5	1, 2, 3, 4	comfffval 16358	. 2 |- O = ( x e. ( B X. B ) , y e. B |-> ( g e. ( ( 2nd ` x ) ( Hom ` C ) y ) , f e. ( ( Hom ` C ) ` x ) |-> ( g ( x .x. y ) f ) ) )
6		comfffval2.h	. . . . 5 |- H = ( Hom f ` C )
7		xp2nd 7199	. . . . . 6 |- ( x e. ( B X. B ) -> ( 2nd ` x ) e. B )
8	7	adantr 481	. . . . 5 |- ( ( x e. ( B X. B ) /\ y e. B ) -> ( 2nd ` x ) e. B )
9		simpr 477	. . . . 5 |- ( ( x e. ( B X. B ) /\ y e. B ) -> y e. B )
10	6, 2, 3, 8, 9	homfval 16352	. . . 4 |- ( ( x e. ( B X. B ) /\ y e. B ) -> ( ( 2nd ` x ) H y ) = ( ( 2nd ` x ) ( Hom ` C ) y ) )
11		xp1st 7198	. . . . . . . 8 |- ( x e. ( B X. B ) -> ( 1st ` x ) e. B )
12	11	adantr 481	. . . . . . 7 |- ( ( x e. ( B X. B ) /\ y e. B ) -> ( 1st ` x ) e. B )
13	6, 2, 3, 12, 8	homfval 16352	. . . . . 6 |- ( ( x e. ( B X. B ) /\ y e. B ) -> ( ( 1st ` x ) H ( 2nd ` x ) ) = ( ( 1st ` x ) ( Hom ` C ) ( 2nd ` x ) ) )
14		df-ov 6653	. . . . . 6 |- ( ( 1st ` x ) H ( 2nd ` x ) ) = ( H ` <. ( 1st ` x ) , ( 2nd ` x ) >. )
15		df-ov 6653	. . . . . 6 |- ( ( 1st ` x ) ( Hom ` C ) ( 2nd ` x ) ) = ( ( Hom ` C ) ` <. ( 1st ` x ) , ( 2nd ` x ) >. )
16	13, 14, 15	3eqtr3g 2679	. . . . 5 |- ( ( x e. ( B X. B ) /\ y e. B ) -> ( H ` <. ( 1st ` x ) , ( 2nd ` x ) >. ) = ( ( Hom ` C ) ` <. ( 1st ` x ) , ( 2nd ` x ) >. ) )
17		1st2nd2 7205	. . . . . . 7 |- ( x e. ( B X. B ) -> x = <. ( 1st ` x ) , ( 2nd ` x ) >. )
18	17	adantr 481	. . . . . 6 |- ( ( x e. ( B X. B ) /\ y e. B ) -> x = <. ( 1st ` x ) , ( 2nd ` x ) >. )
19	18	fveq2d 6195	. . . . 5 |- ( ( x e. ( B X. B ) /\ y e. B ) -> ( H ` x ) = ( H ` <. ( 1st ` x ) , ( 2nd ` x ) >. ) )
20	18	fveq2d 6195	. . . . 5 |- ( ( x e. ( B X. B ) /\ y e. B ) -> ( ( Hom ` C ) ` x ) = ( ( Hom ` C ) ` <. ( 1st ` x ) , ( 2nd ` x ) >. ) )
21	16, 19, 20	3eqtr4d 2666	. . . 4 |- ( ( x e. ( B X. B ) /\ y e. B ) -> ( H ` x ) = ( ( Hom ` C ) ` x ) )
22		eqidd 2623	. . . 4 |- ( ( x e. ( B X. B ) /\ y e. B ) -> ( g ( x .x. y ) f ) = ( g ( x .x. y ) f ) )
23	10, 21, 22	mpt2eq123dv 6717	. . 3 |- ( ( x e. ( B X. B ) /\ y e. B ) -> ( g e. ( ( 2nd ` x ) H y ) , f e. ( H ` x ) |-> ( g ( x .x. y ) f ) ) = ( g e. ( ( 2nd ` x ) ( Hom ` C ) y ) , f e. ( ( Hom ` C ) ` x ) |-> ( g ( x .x. y ) f ) ) )
24	23	mpt2eq3ia 6720	. 2 |- ( 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. ( B X. B ) , y e. B |-> ( g e. ( ( 2nd ` x ) ( Hom ` C ) y ) , f e. ( ( Hom ` C ) ` x ) |-> ( g ( x .x. y ) f ) ) )
25	5, 24	eqtr4i 2647	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:   /\ wa 384   = wceq 1483   e. wcel 1990   <.cop 4183   X. cxp 5112   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   1stc1st 7166   2ndc2nd 7167   Basecbs 15857   Hom chom 15952  compcco 15953   Hom _f chomf 16327  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-homf 16331  df-comf 16332

This theorem is referenced by: (None)