Theorem resflem 5577

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 5577 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 3091	. . . . 5 |- ( ph -> ( x e. A -> x e. V ) )
3		resflem.1	. . . . . . 7 |- ( ph -> F : V --> X )
4		fdm 5273	. . . . . . 7 |- ( F : V --> X -> dom F = V )
5	3, 4	syl 14	. . . . . 6 |- ( ph -> dom F = V )
6	5	eleq2d 2207	. . . . 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 2502	. 2 |- ( ph -> A. x e. A ( x e. dom F /\ ( F ` x ) e. Y ) )
12		ffun 5270	. . . 4 |- ( F : V --> X -> Fun F )
13	3, 12	syl 14	. . 3 |- ( ph -> Fun F )
14		ffvresb 5576	. . 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 1331   e. wcel 1480   A.wral 2414   C_ wss 3066   dom cdm 4534   |` cres 4536   Fun wfun 5112   -->wf 5114   ` cfv 5118

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-sbc 2905  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-mpt 3986  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-fv 5126

This theorem is referenced by: (None)