Theorem resfvresima 5929

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 5700	. . 3 |- ( ph -> ( ( F |` S ) ` X ) = ( F ` X ) )
3	2	fveq2d 5679	. 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 5924	. . . 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 5700	. 2 |- ( ph -> ( ( H |` ( F " S ) ) ` ( F ` X ) ) = ( H ` ( F ` X ) ) )
10	3, 9	eqtrd 2267	1 |- ( ph -> ( ( H |` ( F " S ) ) ` ( ( F |` S ) ` X ) ) = ( H ` ( F ` X ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2205   C_ wss 3214   dom cdm 4754   |` cres 4756   "cima 4757   Fun wfun 5351   ` cfv 5357

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-sbc 3046  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-opab 4177  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-fv 5365

This theorem is referenced by:  wlkres  16503