Theorem funimass4f 24001

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 5439	. . . . 5 |- F/ x Fun F
3		funimass4f.1	. . . . . 6 |- F/_ x A
4	1	nfdm 5074	. . . . . 6 |- F/_ x dom F
5	3, 4	nfss 3305	. . . . 5 |- F/ x A C_ dom F
6	2, 5	nfan 1842	. . . 4 |- F/ x ( Fun F /\ A C_ dom F )
7	1, 3	nfima 5174	. . . . 5 |- F/_ x ( F " A )
8		funimass4f.2	. . . . 5 |- F/_ x B
9	7, 8	nfss 3305	. . . 4 |- F/ x ( F " A ) C_ B
10	6, 9	nfan 1842	. . 3 |- F/ x ( ( Fun F /\ A C_ dom F ) /\ ( F " A ) C_ B )
11		funfvima2 5937	. . . 4 |- ( ( Fun F /\ A C_ dom F ) -> ( x e. A -> ( F ` x ) e. ( F " A ) ) )
12		ssel 3306	. . . 4 |- ( ( F " A ) C_ B -> ( ( F ` x ) e. ( F " A ) -> ( F ` x ) e. B ) )
13	11, 12	sylan9 639	. . 3 |- ( ( ( Fun F /\ A C_ dom F ) /\ ( F " A ) C_ B ) -> ( x e. A -> ( F ` x ) e. B ) )
14	10, 13	ralrimi 2751	. 2 |- ( ( ( Fun F /\ A C_ dom F ) /\ ( F " A ) C_ B ) -> A. x e. A ( F ` x ) e. B )
15	3, 1	dfimafnf 24000	. . . 4 |- ( ( Fun F /\ A C_ dom F ) -> ( F " A ) = { y | E. x e. A y = ( F ` x ) } )
16	15	adantr 452	. . 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 23950	. . . 4 |- ( A. x e. A ( F ` x ) e. B -> { y | E. x e. A y = ( F ` x ) } C_ B )
18	17	adantl 453	. . 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 3346	. 2 |- ( ( ( Fun F /\ A C_ dom F ) /\ A. x e. A ( F ` x ) e. B ) -> ( F " A ) C_ B )
20	14, 19	impbida 806	1 |- ( ( Fun F /\ A C_ dom F ) -> ( ( F " A ) C_ B <-> A. x e. A ( F ` x ) e. B ) )

Syntax hints:   -> wi 4   <-> wb 177   /\ wa 359   = wceq 1649   e. wcel 1721   {cab 2394   F/_wnfc 2531   A.wral 2670   E.wrex 2671   C_ wss 3284   dom cdm 4841   "cima 4844   Fun wfun 5411   ` cfv 5417

This theorem is referenced by:  ballotlem7  24750

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-sep 4294  ax-nul 4302  ax-pr 4367

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-rab 2679  df-v 2922  df-sbc 3126  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-br 4177  df-opab 4231  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-fv 5425