Theorem resfvresima 6494

Description: The value of the function value of a restriction for a function restricted to the image of the restricting subset. (Contributed by AV, 6-Mar-2021.)

Hypotheses

Ref	Expression
resfvresima.f	|- ( ph -> Fun F )
resfvresima.s	|- ( ph -> S C_ dom F )
resfvresima.x	|- ( ph -> X e. S )

Assertion

Ref	Expression
resfvresima	|- ( ph -> ( ( H |` ( F " S ) ) ` ( ( F |` S ) ` X ) ) = ( H ` ( F ` X ) ) )

Proof of Theorem resfvresima

Step	Hyp	Ref	Expression
1		resfvresima.x	. . . 4 |- ( ph -> X e. S )
2		fvres 6207	. . . 4 |- ( X e. S -> ( ( F |` S ) ` X ) = ( F ` X ) )
3	1, 2	syl 17	. . 3 |- ( ph -> ( ( F |` S ) ` X ) = ( F ` X ) )
4	3	fveq2d 6195	. 2 |- ( ph -> ( ( H |` ( F " S ) ) ` ( ( F |` S ) ` X ) ) = ( ( H |` ( F " S ) ) ` ( F ` X ) ) )
5		resfvresima.f	. . . . 5 |- ( ph -> Fun F )
6		resfvresima.s	. . . . 5 |- ( ph -> S C_ dom F )
7	5, 6	jca 554	. . . 4 |- ( ph -> ( Fun F /\ S C_ dom F ) )
8		funfvima2 6493	. . . 4 |- ( ( Fun F /\ S C_ dom F ) -> ( X e. S -> ( F ` X ) e. ( F " S ) ) )
9	7, 1, 8	sylc 65	. . 3 |- ( ph -> ( F ` X ) e. ( F " S ) )
10		fvres 6207	. . 3 |- ( ( F ` X ) e. ( F " S ) -> ( ( H |` ( F " S ) ) ` ( F ` X ) ) = ( H ` ( F ` X ) ) )
11	9, 10	syl 17	. 2 |- ( ph -> ( ( H |` ( F " S ) ) ` ( F ` X ) ) = ( H ` ( F ` X ) ) )
12	4, 11	eqtrd 2656	1 |- ( ph -> ( ( H |` ( F " S ) ) ` ( ( F |` S ) ` X ) ) = ( H ` ( F ` X ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   C_ wss 3574   dom cdm 5114   |` cres 5116   "cima 5117   Fun wfun 5882   ` cfv 5888

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-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-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  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-br 4654  df-opab 4713  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-fv 5896

This theorem is referenced by:  wlkres  26567