Theorem funimass4f 29437

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 5911	. . . . 5 |- F/ x Fun F
3		funimass4f.1	. . . . . 6 |- F/_ x A
4	1	nfdm 5367	. . . . . 6 |- F/_ x dom F
5	3, 4	nfss 3596	. . . . 5 |- F/ x A C_ dom F
6	2, 5	nfan 1828	. . . 4 |- F/ x ( Fun F /\ A C_ dom F )
7	1, 3	nfima 5474	. . . . 5 |- F/_ x ( F " A )
8		funimass4f.2	. . . . 5 |- F/_ x B
9	7, 8	nfss 3596	. . . 4 |- F/ x ( F " A ) C_ B
10	6, 9	nfan 1828	. . 3 |- F/ x ( ( Fun F /\ A C_ dom F ) /\ ( F " A ) C_ B )
11		funfvima2 6493	. . . 4 |- ( ( Fun F /\ A C_ dom F ) -> ( x e. A -> ( F ` x ) e. ( F " A ) ) )
12		ssel 3597	. . . 4 |- ( ( F " A ) C_ B -> ( ( F ` x ) e. ( F " A ) -> ( F ` x ) e. B ) )
13	11, 12	sylan9 689	. . 3 |- ( ( ( Fun F /\ A C_ dom F ) /\ ( F " A ) C_ B ) -> ( x e. A -> ( F ` x ) e. B ) )
14	10, 13	ralrimi 2957	. 2 |- ( ( ( Fun F /\ A C_ dom F ) /\ ( F " A ) C_ B ) -> A. x e. A ( F ` x ) e. B )
15	3, 1	dfimafnf 29436	. . . 4 |- ( ( Fun F /\ A C_ dom F ) -> ( F " A ) = { y | E. x e. A y = ( F ` x ) } )
16	15	adantr 481	. . 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 29350	. . . 4 |- ( A. x e. A ( F ` x ) e. B -> { y | E. x e. A y = ( F ` x ) } C_ B )
18	17	adantl 482	. . 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 3639	. 2 |- ( ( ( Fun F /\ A C_ dom F ) /\ A. x e. A ( F ` x ) e. B ) -> ( F " A ) C_ B )
20	14, 19	impbida 877	1 |- ( ( Fun F /\ A C_ dom F ) -> ( ( F " A ) C_ B <-> A. x e. A ( F ` x ) e. B ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   F/_wnfc 2751   A.wral 2912   E.wrex 2913   C_ wss 3574   dom cdm 5114   "cima 5117   Fun wfun 5882   ` cfv 5888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-fv 5896

This theorem is referenced by:  ballotlem7  30597