Theorem resfvresima 5950

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 5718	. . 3 |- ( ph -> ( ( F |` S ) ` X ) = ( F ` X ) )
3	2	fveq2d 5697	. 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 5945	. . . 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 5718	. 2 |- ( ph -> ( ( H |` ( F " S ) ) ` ( F ` X ) ) = ( H ` ( F ` X ) ) )
10	3, 9	eqtrd 2271	1 |- ( ph -> ( ( H |` ( F " S ) ) ` ( ( F |` S ) ` X ) ) = ( H ` ( F ` X ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   C_ wss 3220   dom cdm 4772   |` cres 4774   "cima 4775   Fun wfun 5369   ` cfv 5375

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-pow 4309  ax-pr 4344

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-br 4129  df-opab 4191  df-id 4436  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-fv 5383

This theorem is referenced by:  wlkres  16603