Theorem resflem 5693

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 5693 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 3166	. . . . 5 |- ( ph -> ( x e. A -> x e. V ) )
3		resflem.1	. . . . . . 7 |- ( ph -> F : V --> X )
4		fdm 5383	. . . . . . 7 |- ( F : V --> X -> dom F = V )
5	3, 4	syl 14	. . . . . 6 |- ( ph -> dom F = V )
6	5	eleq2d 2257	. . . . 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 2559	. 2 |- ( ph -> A. x e. A ( x e. dom F /\ ( F ` x ) e. Y ) )
12		ffun 5380	. . . 4 |- ( F : V --> X -> Fun F )
13	3, 12	syl 14	. . 3 |- ( ph -> Fun F )
14		ffvresb 5692	. . 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 1363   e. wcel 2158   A.wral 2465   C_ wss 3141   dom cdm 4638   |` cres 4640   Fun wfun 5222   -->wf 5224   ` cfv 5228

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 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-v 2751  df-sbc 2975  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-br 4016  df-opab 4077  df-mpt 4078  df-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-fv 5236

This theorem is referenced by:  bj-charfun  14912