Theorem funimass4f 24049

Description: Membership relation for the values of a function whose image is a subclass. (Contributed by Thierry Arnoux, 24-Apr-2017.)

Hypotheses

Ref	Expression
funimass4f.1	|- F/_ x A
funimass4f.2	|- F/_ x B
funimass4f.3	|- F/_ x F

Assertion

Ref	Expression
funimass4f	|- ( ( Fun F /\ A C_ dom F ) -> ( ( F " A ) C_ B <-> A. x e. A ( F ` x ) e. B ) )

Proof of Theorem funimass4f

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		funimass4f.3	. . . . . 6 |- F/_ x F
2	1	nffun 5479	. . . . 5 |- F/ x Fun F
3		funimass4f.1	. . . . . 6 |- F/_ x A
4	1	nfdm 5114	. . . . . 6 |- F/_ x dom F
5	3, 4	nfss 3343	. . . . 5 |- F/ x A C_ dom F
6	2, 5	nfan 1847	. . . 4 |- F/ x ( Fun F /\ A C_ dom F )
7	1, 3	nfima 5214	. . . . 5 |- F/_ x ( F " A )
8		funimass4f.2	. . . . 5 |- F/_ x B
9	7, 8	nfss 3343	. . . 4 |- F/ x ( F " A ) C_ B
10	6, 9	nfan 1847	. . 3 |- F/ x ( ( Fun F /\ A C_ dom F ) /\ ( F " A ) C_ B )
11		funfvima2 5977	. . . 4 |- ( ( Fun F /\ A C_ dom F ) -> ( x e. A -> ( F ` x ) e. ( F " A ) ) )
12		ssel 3344	. . . 4 |- ( ( F " A ) C_ B -> ( ( F ` x ) e. ( F " A ) -> ( F ` x ) e. B ) )
13	11, 12	sylan9 640	. . 3 |- ( ( ( Fun F /\ A C_ dom F ) /\ ( F " A ) C_ B ) -> ( x e. A -> ( F ` x ) e. B ) )
14	10, 13	ralrimi 2789	. 2 |- ( ( ( Fun F /\ A C_ dom F ) /\ ( F " A ) C_ B ) -> A. x e. A ( F ` x ) e. B )
15	3, 1	dfimafnf 24048	. . . 4 |- ( ( Fun F /\ A C_ dom F ) -> ( F " A ) = { y | E. x e. A y = ( F ` x ) } )
16	15	adantr 453	. . 3 |- ( ( ( Fun F /\ A C_ dom F ) /\ A. x e. A ( F ` x ) e. B ) -> ( F " A ) = { y | E. x e. A y = ( F ` x ) } )
17	8	abrexss 23998	. . . 4 |- ( A. x e. A ( F ` x ) e. B -> { y | E. x e. A y = ( F ` x ) } C_ B )
18	17	adantl 454	. . 3 |- ( ( ( Fun F /\ A C_ dom F ) /\ A. x e. A ( F ` x ) e. B ) -> { y | E. x e. A y = ( F ` x ) } C_ B )
19	16, 18	eqsstrd 3384	. 2 |- ( ( ( Fun F /\ A C_ dom F ) /\ A. x e. A ( F ` x ) e. B ) -> ( F " A ) C_ B )
20	14, 19	impbida 807	1 |- ( ( Fun F /\ A C_ dom F ) -> ( ( F " A ) C_ B <-> A. x e. A ( F ` x ) e. B ) )

Syntax hints:   -> wi 4   <-> wb 178   /\ wa 360   = wceq 1653   e. wcel 1726   {cab 2424   F/_wnfc 2561   A.wral 2707   E.wrex 2708   C_ wss 3322   dom cdm 4881   "cima 4884   Fun wfun 5451   ` cfv 5457

This theorem is referenced by:  ballotlem7  24798

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406

This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-fv 5465