Theorem ofco 6578

Description: The composition of a function operation with another function. (Contributed by Mario Carneiro, 19-Dec-2014.)

Hypotheses

Ref	Expression
ofco.1	|- ( ph -> F Fn A )
ofco.2	|- ( ph -> G Fn B )
ofco.3	|- ( ph -> H : D --> C )
ofco.4	|- ( ph -> A e. V )
ofco.5	|- ( ph -> B e. W )
ofco.6	|- ( ph -> D e. X )
ofco.7	|- ( A i^i B ) = C

Assertion

Ref	Expression
ofco	|- ( ph -> ( ( F oF R G ) o. H ) = ( ( F o. H ) oF R ( G o. H ) ) )

Proof of Theorem ofco

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

Step	Hyp	Ref	Expression
1		ofco.3	. . . 4 |- ( ph -> H : D --> C )
2	1	ffvelrnda 6045	. . 3 |- ( ( ph /\ x e. D ) -> ( H ` x ) e. C )
3	1	feqmptd 5941	. . 3 |- ( ph -> H = ( x e. D |-> ( H ` x ) ) )
4		ofco.1	. . . 4 |- ( ph -> F Fn A )
5		ofco.2	. . . 4 |- ( ph -> G Fn B )
6		ofco.4	. . . 4 |- ( ph -> A e. V )
7		ofco.5	. . . 4 |- ( ph -> B e. W )
8		ofco.7	. . . 4 |- ( A i^i B ) = C
9		eqidd 2463	. . . 4 |- ( ( ph /\ y e. A ) -> ( F ` y ) = ( F ` y ) )
10		eqidd 2463	. . . 4 |- ( ( ph /\ y e. B ) -> ( G ` y ) = ( G ` y ) )
11	4, 5, 6, 7, 8, 9, 10	offval 6565	. . 3 |- ( ph -> ( F oF R G ) = ( y e. C |-> ( ( F ` y ) R ( G ` y ) ) ) )
12		fveq2 5888	. . . 4 |- ( y = ( H ` x ) -> ( F ` y ) = ( F ` ( H ` x ) ) )
13		fveq2 5888	. . . 4 |- ( y = ( H ` x ) -> ( G ` y ) = ( G ` ( H ` x ) ) )
14	12, 13	oveq12d 6333	. . 3 |- ( y = ( H ` x ) -> ( ( F ` y ) R ( G ` y ) ) = ( ( F ` ( H ` x ) ) R ( G ` ( H ` x ) ) ) )
15	2, 3, 11, 14	fmptco 6080	. 2 |- ( ph -> ( ( F oF R G ) o. H ) = ( x e. D |-> ( ( F ` ( H ` x ) ) R ( G ` ( H ` x ) ) ) ) )
16		inss1 3664	. . . . . 6 |- ( A i^i B ) C_ A
17	8, 16	eqsstr3i 3475	. . . . 5 |- C C_ A
18		fss 5760	. . . . 5 |- ( ( H : D --> C /\ C C_ A ) -> H : D --> A )
19	1, 17, 18	sylancl 673	. . . 4 |- ( ph -> H : D --> A )
20		fnfco 5771	. . . 4 |- ( ( F Fn A /\ H : D --> A ) -> ( F o. H ) Fn D )
21	4, 19, 20	syl2anc 671	. . 3 |- ( ph -> ( F o. H ) Fn D )
22		inss2 3665	. . . . . 6 |- ( A i^i B ) C_ B
23	8, 22	eqsstr3i 3475	. . . . 5 |- C C_ B
24		fss 5760	. . . . 5 |- ( ( H : D --> C /\ C C_ B ) -> H : D --> B )
25	1, 23, 24	sylancl 673	. . . 4 |- ( ph -> H : D --> B )
26		fnfco 5771	. . . 4 |- ( ( G Fn B /\ H : D --> B ) -> ( G o. H ) Fn D )
27	5, 25, 26	syl2anc 671	. . 3 |- ( ph -> ( G o. H ) Fn D )
28		ofco.6	. . 3 |- ( ph -> D e. X )
29		inidm 3653	. . 3 |- ( D i^i D ) = D
30		ffn 5751	. . . . 5 |- ( H : D --> C -> H Fn D )
31	1, 30	syl 17	. . . 4 |- ( ph -> H Fn D )
32		fvco2 5963	. . . 4 |- ( ( H Fn D /\ x e. D ) -> ( ( F o. H ) ` x ) = ( F ` ( H ` x ) ) )
33	31, 32	sylan 478	. . 3 |- ( ( ph /\ x e. D ) -> ( ( F o. H ) ` x ) = ( F ` ( H ` x ) ) )
34		fvco2 5963	. . . 4 |- ( ( H Fn D /\ x e. D ) -> ( ( G o. H ) ` x ) = ( G ` ( H ` x ) ) )
35	31, 34	sylan 478	. . 3 |- ( ( ph /\ x e. D ) -> ( ( G o. H ) ` x ) = ( G ` ( H ` x ) ) )
36	21, 27, 28, 28, 29, 33, 35	offval 6565	. 2 |- ( ph -> ( ( F o. H ) oF R ( G o. H ) ) = ( x e. D |-> ( ( F ` ( H ` x ) ) R ( G ` ( H ` x ) ) ) ) )
37	15, 36	eqtr4d 2499	1 |- ( ph -> ( ( F oF R G ) o. H ) = ( ( F o. H ) oF R ( G o. H ) ) )

Syntax hints:   -> wi 4   /\ wa 375   = wceq 1455   e. wcel 1898   i^i cin 3415   C_ wss 3416   |-> cmpt 4475   o. ccom 4857   Fn wfn 5596   -->wf 5597   ` cfv 5601  (class class class)co 6315   oFcof 6556

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-rep 4529  ax-sep 4539  ax-nul 4548  ax-pow 4595  ax-pr 4653

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-reu 2756  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4213  df-iun 4294  df-br 4417  df-opab 4476  df-mpt 4477  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-ov 6318  df-oprab 6319  df-mpt2 6320  df-of 6558

This theorem is referenced by:  gsumzaddlem  17603  coe1add  18906  pf1ind  18992  mendring  36103