Theorem comffval 16359

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 )
comffval.x	|- ( ph -> X e. B )
comffval.y	|- ( ph -> Y e. B )
comffval.z	|- ( ph -> Z e. B )

Assertion

Ref	Expression
comffval	|- ( ph -> ( <. X , Y >. O Z ) = ( g e. ( Y H Z ) , f e. ( X H Y ) |-> ( g ( <. X , Y >. .x. Z ) f ) ) )

Distinct variable groups:   f, g, C   ph, f, g   .x. , f, g   f, X, g   f, Y, g   f, Z, g   f, H, g

Allowed substitution hints:   B( f, g)   O( f, g)

Proof of Theorem comffval

Dummy variables x z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		comfffval.o	. . . 4 |- O = (compf ` C )
2		comfffval.b	. . . 4 |- B = ( Base ` C )
3		comfffval.h	. . . 4 |- H = ( Hom ` C )
4		comfffval.x	. . . 4 |- .x. = (comp ` C )
5	1, 2, 3, 4	comfffval 16358	. . 3 |- O = ( x e. ( B X. B ) , z e. B |-> ( g e. ( ( 2nd ` x ) H z ) , f e. ( H ` x ) |-> ( g ( x .x. z ) f ) ) )
6	5	a1i 11	. 2 |- ( ph -> O = ( x e. ( B X. B ) , z e. B |-> ( g e. ( ( 2nd ` x ) H z ) , f e. ( H ` x ) |-> ( g ( x .x. z ) f ) ) ) )
7		simprl 794	. . . . . 6 |- ( ( ph /\ ( x = <. X , Y >. /\ z = Z ) ) -> x = <. X , Y >. )
8	7	fveq2d 6195	. . . . 5 |- ( ( ph /\ ( x = <. X , Y >. /\ z = Z ) ) -> ( 2nd ` x ) = ( 2nd ` <. X , Y >. ) )
9		comffval.x	. . . . . . 7 |- ( ph -> X e. B )
10		comffval.y	. . . . . . 7 |- ( ph -> Y e. B )
11		op2ndg 7181	. . . . . . 7 |- ( ( X e. B /\ Y e. B ) -> ( 2nd ` <. X , Y >. ) = Y )
12	9, 10, 11	syl2anc 693	. . . . . 6 |- ( ph -> ( 2nd ` <. X , Y >. ) = Y )
13	12	adantr 481	. . . . 5 |- ( ( ph /\ ( x = <. X , Y >. /\ z = Z ) ) -> ( 2nd ` <. X , Y >. ) = Y )
14	8, 13	eqtrd 2656	. . . 4 |- ( ( ph /\ ( x = <. X , Y >. /\ z = Z ) ) -> ( 2nd ` x ) = Y )
15		simprr 796	. . . 4 |- ( ( ph /\ ( x = <. X , Y >. /\ z = Z ) ) -> z = Z )
16	14, 15	oveq12d 6668	. . 3 |- ( ( ph /\ ( x = <. X , Y >. /\ z = Z ) ) -> ( ( 2nd ` x ) H z ) = ( Y H Z ) )
17	7	fveq2d 6195	. . . 4 |- ( ( ph /\ ( x = <. X , Y >. /\ z = Z ) ) -> ( H ` x ) = ( H ` <. X , Y >. ) )
18		df-ov 6653	. . . 4 |- ( X H Y ) = ( H ` <. X , Y >. )
19	17, 18	syl6eqr 2674	. . 3 |- ( ( ph /\ ( x = <. X , Y >. /\ z = Z ) ) -> ( H ` x ) = ( X H Y ) )
20	7, 15	oveq12d 6668	. . . 4 |- ( ( ph /\ ( x = <. X , Y >. /\ z = Z ) ) -> ( x .x. z ) = ( <. X , Y >. .x. Z ) )
21	20	oveqd 6667	. . 3 |- ( ( ph /\ ( x = <. X , Y >. /\ z = Z ) ) -> ( g ( x .x. z ) f ) = ( g ( <. X , Y >. .x. Z ) f ) )
22	16, 19, 21	mpt2eq123dv 6717	. 2 |- ( ( ph /\ ( x = <. X , Y >. /\ z = Z ) ) -> ( g e. ( ( 2nd ` x ) H z ) , f e. ( H ` x ) |-> ( g ( x .x. z ) f ) ) = ( g e. ( Y H Z ) , f e. ( X H Y ) |-> ( g ( <. X , Y >. .x. Z ) f ) ) )
23		opelxpi 5148	. . 3 |- ( ( X e. B /\ Y e. B ) -> <. X , Y >. e. ( B X. B ) )
24	9, 10, 23	syl2anc 693	. 2 |- ( ph -> <. X , Y >. e. ( B X. B ) )
25		comffval.z	. 2 |- ( ph -> Z e. B )
26		ovex 6678	. . . 4 |- ( Y H Z ) e. _V
27		ovex 6678	. . . 4 |- ( X H Y ) e. _V
28	26, 27	mpt2ex 7247	. . 3 |- ( g e. ( Y H Z ) , f e. ( X H Y ) |-> ( g ( <. X , Y >. .x. Z ) f ) ) e. _V
29	28	a1i 11	. 2 |- ( ph -> ( g e. ( Y H Z ) , f e. ( X H Y ) |-> ( g ( <. X , Y >. .x. Z ) f ) ) e. _V )
30	6, 22, 24, 25, 29	ovmpt2d 6788	1 |- ( ph -> ( <. X , Y >. O Z ) = ( g e. ( Y H Z ) , f e. ( X H Y ) |-> ( g ( <. X , Y >. .x. Z ) f ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   <.cop 4183   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:  comfval  16360  comffval2  16362  comffn  16365