Theorem resfvresima 5880

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 5654	. . 3 |- ( ph -> ( ( F |` S ) ` X ) = ( F ` X ) )
3	2	fveq2d 5633	. 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 5876	. . . 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 5654	. 2 |- ( ph -> ( ( H |` ( F " S ) ) ` ( F ` X ) ) = ( H ` ( F ` X ) ) )
10	3, 9	eqtrd 2262	1 |- ( ph -> ( ( H |` ( F " S ) ) ` ( ( F |` S ) ` X ) ) = ( H ` ( F ` X ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   e. wcel 2200   C_ wss 3197   dom cdm 4719   |` cres 4721   "cima 4722   Fun wfun 5312   ` cfv 5318

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-sbc 3029  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-fv 5326

This theorem is referenced by:  wlkres  16098