Theorem resflem 5584

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 5584 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 3096	. . . . 5 |- ( ph -> ( x e. A -> x e. V ) )
3		resflem.1	. . . . . . 7 |- ( ph -> F : V --> X )
4		fdm 5278	. . . . . . 7 |- ( F : V --> X -> dom F = V )
5	3, 4	syl 14	. . . . . 6 |- ( ph -> dom F = V )
6	5	eleq2d 2209	. . . . 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 2504	. 2 |- ( ph -> A. x e. A ( x e. dom F /\ ( F ` x ) e. Y ) )
12		ffun 5275	. . . 4 |- ( F : V --> X -> Fun F )
13	3, 12	syl 14	. . 3 |- ( ph -> Fun F )
14		ffvresb 5583	. . 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 2416   C_ wss 3071   dom cdm 4539   |` cres 4541   Fun wfun 5117   -->wf 5119   ` cfv 5123

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 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-sbc 2910  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-fv 5131

This theorem is referenced by: (None)