Theorem resfvresima 5896

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	1	fvresd 5667	. . 3 |- ( ph -> ( ( F |` S ) ` X ) = ( F ` X ) )
3	2	fveq2d 5646	. 2 |- ( ph -> ( ( H |` ( F " S ) ) ` ( ( F |` S ) ` X ) ) = ( ( H |` ( F " S ) ) ` ( F ` X ) ) )
4		resfvresima.f	. . . . 5 |- ( ph -> Fun F )
5		resfvresima.s	. . . . 5 |- ( ph -> S C_ dom F )
6	4, 5	jca 306	. . . 4 |- ( ph -> ( Fun F /\ S C_ dom F ) )
7		funfvima2 5892	. . . 4 |- ( ( Fun F /\ S C_ dom F ) -> ( X e. S -> ( F ` X ) e. ( F " S ) ) )
8	6, 1, 7	sylc 62	. . 3 |- ( ph -> ( F ` X ) e. ( F " S ) )
9	8	fvresd 5667	. 2 |- ( ph -> ( ( H |` ( F " S ) ) ` ( F ` X ) ) = ( H ` ( F ` X ) ) )
10	3, 9	eqtrd 2263	1 |- ( ph -> ( ( H |` ( F " S ) ) ` ( ( F |` S ) ` X ) ) = ( H ` ( F ` X ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1397   e. wcel 2201   C_ wss 3199   dom cdm 4727   |` cres 4729   "cima 4730   Fun wfun 5322   ` cfv 5328

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2204  ax-ext 2212  ax-sep 4208  ax-pow 4266  ax-pr 4301

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1810  df-eu 2081  df-mo 2082  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ral 2514  df-rex 2515  df-v 2803  df-sbc 3031  df-un 3203  df-in 3205  df-ss 3212  df-pw 3655  df-sn 3676  df-pr 3677  df-op 3679  df-uni 3895  df-br 4090  df-opab 4152  df-id 4392  df-xp 4733  df-rel 4734  df-cnv 4735  df-co 4736  df-dm 4737  df-rn 4738  df-res 4739  df-ima 4740  df-iota 5288  df-fun 5330  df-fn 5331  df-fv 5336

This theorem is referenced by:  wlkres  16259