Theorem resflem 5811

Description: A lemma to bound the range of a restriction. The conclusion would also hold with ( X i^i Y ) in place of Y (provided x does not occur in X). If that stronger result is needed, it is however simpler to use the instance of resflem 5811 where ( X i^i Y ) is substituted for Y (in both the conclusion and the third hypothesis). (Contributed by BJ, 4-Jul-2022.)

Hypotheses

Ref	Expression
resflem.1	|- ( ph -> F : V --> X )
resflem.2	|- ( ph -> A C_ V )
resflem.3	|- ( ( ph /\ x e. A ) -> ( F ` x ) e. Y )

Assertion

Ref	Expression
resflem	|- ( ph -> ( F |` A ) : A --> Y )

Distinct variable groups:   x, A   ph, x   x, F   x, Y

Allowed substitution hints:   V( x)   X( x)

Proof of Theorem resflem

Step	Hyp	Ref	Expression
1		resflem.2	. . . . . 6 |- ( ph -> A C_ V )
2	1	sseld 3226	. . . . 5 |- ( ph -> ( x e. A -> x e. V ) )
3		resflem.1	. . . . . . 7 |- ( ph -> F : V --> X )
4		fdm 5488	. . . . . . 7 |- ( F : V --> X -> dom F = V )
5	3, 4	syl 14	. . . . . 6 |- ( ph -> dom F = V )
6	5	eleq2d 2301	. . . . 5 |- ( ph -> ( x e. dom F <-> x e. V ) )
7	2, 6	sylibrd 169	. . . 4 |- ( ph -> ( x e. A -> x e. dom F ) )
8		resflem.3	. . . . 5 |- ( ( ph /\ x e. A ) -> ( F ` x ) e. Y )
9	8	ex 115	. . . 4 |- ( ph -> ( x e. A -> ( F ` x ) e. Y ) )
10	7, 9	jcad 307	. . 3 |- ( ph -> ( x e. A -> ( x e. dom F /\ ( F ` x ) e. Y ) ) )
11	10	ralrimiv 2604	. 2 |- ( ph -> A. x e. A ( x e. dom F /\ ( F ` x ) e. Y ) )
12		ffun 5485	. . . 4 |- ( F : V --> X -> Fun F )
13	3, 12	syl 14	. . 3 |- ( ph -> Fun F )
14		ffvresb 5810	. . 3 |- ( Fun F -> ( ( F |` A ) : A --> Y <-> A. x e. A ( x e. dom F /\ ( F ` x ) e. Y ) ) )
15	13, 14	syl 14	. 2 |- ( ph -> ( ( F |` A ) : A --> Y <-> A. x e. A ( x e. dom F /\ ( F ` x ) e. Y ) ) )
16	11, 15	mpbird 167	1 |- ( ph -> ( F |` A ) : A --> Y )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1397   e. wcel 2202   A.wral 2510   C_ wss 3200   dom cdm 4725   |` cres 4727   Fun wfun 5320   -->wf 5322   ` cfv 5326

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 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-sbc 3032  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-fv 5334

This theorem is referenced by:  bj-charfun  16402