Theorem ofoprabco 29464

Description: Function operation as a composition with an operation. (Contributed by Thierry Arnoux, 4-Jun-2017.)

Hypotheses

Ref	Expression
ofoprabco.1	|- F/_ a M
ofoprabco.2	|- ( ph -> F : A --> B )
ofoprabco.3	|- ( ph -> G : A --> C )
ofoprabco.4	|- ( ph -> A e. V )
ofoprabco.5	|- ( ph -> M = ( a e. A |-> <. ( F ` a ) , ( G ` a ) >. ) )
ofoprabco.6	|- ( ph -> N = ( x e. B , y e. C |-> ( x R y ) ) )

Assertion

Ref	Expression
ofoprabco	|- ( ph -> ( F oF R G ) = ( N o. M ) )

Distinct variable groups:   x, a, y, A   B, a, x, y   C, a, x, y   F, a, x, y   G, a, x, y   N, a   R, a, x, y   ph, a, x, y

Allowed substitution hints:   M( x, y, a)   N( x, y)   V( x, y, a)

Proof of Theorem ofoprabco

Step	Hyp	Ref	Expression
1		ofoprabco.5	. . . . . 6 |- ( ph -> M = ( a e. A |-> <. ( F ` a ) , ( G ` a ) >. ) )
2		ofoprabco.2	. . . . . . . 8 |- ( ph -> F : A --> B )
3	2	ffvelrnda 6359	. . . . . . 7 |- ( ( ph /\ a e. A ) -> ( F ` a ) e. B )
4		ofoprabco.3	. . . . . . . 8 |- ( ph -> G : A --> C )
5	4	ffvelrnda 6359	. . . . . . 7 |- ( ( ph /\ a e. A ) -> ( G ` a ) e. C )
6		opelxpi 5148	. . . . . . 7 |- ( ( ( F ` a ) e. B /\ ( G ` a ) e. C ) -> <. ( F ` a ) , ( G ` a ) >. e. ( B X. C ) )
7	3, 5, 6	syl2anc 693	. . . . . 6 |- ( ( ph /\ a e. A ) -> <. ( F ` a ) , ( G ` a ) >. e. ( B X. C ) )
8	1, 7	fvmpt2d 6293	. . . . 5 |- ( ( ph /\ a e. A ) -> ( M ` a ) = <. ( F ` a ) , ( G ` a ) >. )
9	8	fveq2d 6195	. . . 4 |- ( ( ph /\ a e. A ) -> ( N ` ( M ` a ) ) = ( N ` <. ( F ` a ) , ( G ` a ) >. ) )
10		df-ov 6653	. . . . 5 |- ( ( F ` a ) N ( G ` a ) ) = ( N ` <. ( F ` a ) , ( G ` a ) >. )
11	10	a1i 11	. . . 4 |- ( ( ph /\ a e. A ) -> ( ( F ` a ) N ( G ` a ) ) = ( N ` <. ( F ` a ) , ( G ` a ) >. ) )
12		ofoprabco.6	. . . . . 6 |- ( ph -> N = ( x e. B , y e. C |-> ( x R y ) ) )
13	12	adantr 481	. . . . 5 |- ( ( ph /\ a e. A ) -> N = ( x e. B , y e. C |-> ( x R y ) ) )
14		simprl 794	. . . . . 6 |- ( ( ( ph /\ a e. A ) /\ ( x = ( F ` a ) /\ y = ( G ` a ) ) ) -> x = ( F ` a ) )
15		simprr 796	. . . . . 6 |- ( ( ( ph /\ a e. A ) /\ ( x = ( F ` a ) /\ y = ( G ` a ) ) ) -> y = ( G ` a ) )
16	14, 15	oveq12d 6668	. . . . 5 |- ( ( ( ph /\ a e. A ) /\ ( x = ( F ` a ) /\ y = ( G ` a ) ) ) -> ( x R y ) = ( ( F ` a ) R ( G ` a ) ) )
17		ovexd 6680	. . . . 5 |- ( ( ph /\ a e. A ) -> ( ( F ` a ) R ( G ` a ) ) e. _V )
18	13, 16, 3, 5, 17	ovmpt2d 6788	. . . 4 |- ( ( ph /\ a e. A ) -> ( ( F ` a ) N ( G ` a ) ) = ( ( F ` a ) R ( G ` a ) ) )
19	9, 11, 18	3eqtr2d 2662	. . 3 |- ( ( ph /\ a e. A ) -> ( N ` ( M ` a ) ) = ( ( F ` a ) R ( G ` a ) ) )
20	19	mpteq2dva 4744	. 2 |- ( ph -> ( a e. A |-> ( N ` ( M ` a ) ) ) = ( a e. A |-> ( ( F ` a ) R ( G ` a ) ) ) )
21		ovex 6678	. . . . . 6 |- ( x R y ) e. _V
22	21	rgen2w 2925	. . . . 5 |- A. x e. B A. y e. C ( x R y ) e. _V
23		eqid 2622	. . . . . 6 |- ( x e. B , y e. C |-> ( x R y ) ) = ( x e. B , y e. C |-> ( x R y ) )
24	23	fmpt2 7237	. . . . 5 |- ( A. x e. B A. y e. C ( x R y ) e. _V <-> ( x e. B , y e. C |-> ( x R y ) ) : ( B X. C ) --> _V )
25	22, 24	mpbi 220	. . . 4 |- ( x e. B , y e. C |-> ( x R y ) ) : ( B X. C ) --> _V
26	12	feq1d 6030	. . . 4 |- ( ph -> ( N : ( B X. C ) --> _V <-> ( x e. B , y e. C |-> ( x R y ) ) : ( B X. C ) --> _V ) )
27	25, 26	mpbiri 248	. . 3 |- ( ph -> N : ( B X. C ) --> _V )
28	1, 7	fmpt3d 6386	. . 3 |- ( ph -> M : A --> ( B X. C ) )
29		ofoprabco.1	. . . 4 |- F/_ a M
30	29	fcomptf 29458	. . 3 |- ( ( N : ( B X. C ) --> _V /\ M : A --> ( B X. C ) ) -> ( N o. M ) = ( a e. A |-> ( N ` ( M ` a ) ) ) )
31	27, 28, 30	syl2anc 693	. 2 |- ( ph -> ( N o. M ) = ( a e. A |-> ( N ` ( M ` a ) ) ) )
32		ofoprabco.4	. . 3 |- ( ph -> A e. V )
33	2	feqmptd 6249	. . 3 |- ( ph -> F = ( a e. A |-> ( F ` a ) ) )
34	4	feqmptd 6249	. . 3 |- ( ph -> G = ( a e. A |-> ( G ` a ) ) )
35	32, 3, 5, 33, 34	offval2 6914	. 2 |- ( ph -> ( F oF R G ) = ( a e. A |-> ( ( F ` a ) R ( G ` a ) ) ) )
36	20, 31, 35	3eqtr4rd 2667	1 |- ( ph -> ( F oF R G ) = ( N o. M ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   F/_wnfc 2751   A.wral 2912   _Vcvv 3200   <.cop 4183   |-> cmpt 4729   X. cxp 5112   o. ccom 5118   -->wf 5884   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   oFcof 6895

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-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-of 6897  df-1st 7168  df-2nd 7169

This theorem is referenced by:  ofpreima  29465  rrvadd  30514