Theorem resflem 5649

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 5649 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 3141	. . . . 5 |- ( ph -> ( x e. A -> x e. V ) )
3		resflem.1	. . . . . . 7 |- ( ph -> F : V --> X )
4		fdm 5343	. . . . . . 7 |- ( F : V --> X -> dom F = V )
5	3, 4	syl 14	. . . . . 6 |- ( ph -> dom F = V )
6	5	eleq2d 2236	. . . . 5 |- ( ph -> ( x e. dom F <-> x e. V ) )
7	2, 6	sylibrd 168	. . . 4 |- ( ph -> ( x e. A -> x e. dom F ) )
8		resflem.3	. . . . 5 |- ( ( ph /\ x e. A ) -> ( F ` x ) e. Y )
9	8	ex 114	. . . 4 |- ( ph -> ( x e. A -> ( F ` x ) e. Y ) )
10	7, 9	jcad 305	. . 3 |- ( ph -> ( x e. A -> ( x e. dom F /\ ( F ` x ) e. Y ) ) )
11	10	ralrimiv 2538	. 2 |- ( ph -> A. x e. A ( x e. dom F /\ ( F ` x ) e. Y ) )
12		ffun 5340	. . . 4 |- ( F : V --> X -> Fun F )
13	3, 12	syl 14	. . 3 |- ( ph -> Fun F )
14		ffvresb 5648	. . 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 166	1 |- ( ph -> ( F |` A ) : A --> Y )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1343   e. wcel 2136   A.wral 2444   C_ wss 3116   dom cdm 4604   |` cres 4606   Fun wfun 5182   -->wf 5184   ` cfv 5188

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-sbc 2952  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-fv 5196

This theorem is referenced by:  bj-charfun  13689