Theorem ofresid 29444

Description: Applying an operation restricted to the range of the functions does not change the function operation. (Contributed by Thierry Arnoux, 14-Feb-2018.)

Hypotheses

Ref	Expression
ofresid.1	|- ( ph -> F : A --> B )
ofresid.2	|- ( ph -> G : A --> B )
ofresid.3	|- ( ph -> A e. V )

Assertion

Ref	Expression
ofresid	|- ( ph -> ( F oF R G ) = ( F oF ( R |` ( B X. B ) ) G ) )

Proof of Theorem ofresid

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ofresid.1	. . . . . . . 8 |- ( ph -> F : A --> B )
2	1	ffvelrnda 6359	. . . . . . 7 |- ( ( ph /\ x e. A ) -> ( F ` x ) e. B )
3		ofresid.2	. . . . . . . 8 |- ( ph -> G : A --> B )
4	3	ffvelrnda 6359	. . . . . . 7 |- ( ( ph /\ x e. A ) -> ( G ` x ) e. B )
5		opelxp 5146	. . . . . . 7 |- ( <. ( F ` x ) , ( G ` x ) >. e. ( B X. B ) <-> ( ( F ` x ) e. B /\ ( G ` x ) e. B ) )
6	2, 4, 5	sylanbrc 698	. . . . . 6 |- ( ( ph /\ x e. A ) -> <. ( F ` x ) , ( G ` x ) >. e. ( B X. B ) )
7		fvres 6207	. . . . . 6 |- ( <. ( F ` x ) , ( G ` x ) >. e. ( B X. B ) -> ( ( R |` ( B X. B ) ) ` <. ( F ` x ) , ( G ` x ) >. ) = ( R ` <. ( F ` x ) , ( G ` x ) >. ) )
8	6, 7	syl 17	. . . . 5 |- ( ( ph /\ x e. A ) -> ( ( R |` ( B X. B ) ) ` <. ( F ` x ) , ( G ` x ) >. ) = ( R ` <. ( F ` x ) , ( G ` x ) >. ) )
9	8	eqcomd 2628	. . . 4 |- ( ( ph /\ x e. A ) -> ( R ` <. ( F ` x ) , ( G ` x ) >. ) = ( ( R |` ( B X. B ) ) ` <. ( F ` x ) , ( G ` x ) >. ) )
10		df-ov 6653	. . . 4 |- ( ( F ` x ) R ( G ` x ) ) = ( R ` <. ( F ` x ) , ( G ` x ) >. )
11		df-ov 6653	. . . 4 |- ( ( F ` x ) ( R |` ( B X. B ) ) ( G ` x ) ) = ( ( R |` ( B X. B ) ) ` <. ( F ` x ) , ( G ` x ) >. )
12	9, 10, 11	3eqtr4g 2681	. . 3 |- ( ( ph /\ x e. A ) -> ( ( F ` x ) R ( G ` x ) ) = ( ( F ` x ) ( R |` ( B X. B ) ) ( G ` x ) ) )
13	12	mpteq2dva 4744	. 2 |- ( ph -> ( x e. A |-> ( ( F ` x ) R ( G ` x ) ) ) = ( x e. A |-> ( ( F ` x ) ( R |` ( B X. B ) ) ( G ` x ) ) ) )
14		ffn 6045	. . . 4 |- ( F : A --> B -> F Fn A )
15	1, 14	syl 17	. . 3 |- ( ph -> F Fn A )
16		ffn 6045	. . . 4 |- ( G : A --> B -> G Fn A )
17	3, 16	syl 17	. . 3 |- ( ph -> G Fn A )
18		ofresid.3	. . 3 |- ( ph -> A e. V )
19		inidm 3822	. . 3 |- ( A i^i A ) = A
20		eqidd 2623	. . 3 |- ( ( ph /\ x e. A ) -> ( F ` x ) = ( F ` x ) )
21		eqidd 2623	. . 3 |- ( ( ph /\ x e. A ) -> ( G ` x ) = ( G ` x ) )
22	15, 17, 18, 18, 19, 20, 21	offval 6904	. 2 |- ( ph -> ( F oF R G ) = ( x e. A |-> ( ( F ` x ) R ( G ` x ) ) ) )
23	15, 17, 18, 18, 19, 20, 21	offval 6904	. 2 |- ( ph -> ( F oF ( R |` ( B X. B ) ) G ) = ( x e. A |-> ( ( F ` x ) ( R |` ( B X. B ) ) ( G ` x ) ) ) )
24	13, 22, 23	3eqtr4d 2666	1 |- ( ph -> ( F oF R G ) = ( F oF ( R |` ( B X. B ) ) G ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   <.cop 4183   |-> cmpt 4729   X. cxp 5112   |` cres 5116   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   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-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-pr 4906

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

This theorem is referenced by:  sitmcl  30413