Theorem funimass4f 6359

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 5400	. . . . 5 |- F/ x Fun F
3		funimass4f.1	. . . . . 6 |- F/_ x A
4	1	nfdm 5026	. . . . . 6 |- F/_ x dom F
5	3, 4	nfss 3241	. . . . 5 |- F/ x A C_ dom F
6	2, 5	nfan 1618	. . . 4 |- F/ x ( Fun F /\ A C_ dom F )
7	1, 3	nfima 5134	. . . . 5 |- F/_ x ( F " A )
8		funimass4f.2	. . . . 5 |- F/_ x B
9	7, 8	nfss 3241	. . . 4 |- F/ x ( F " A ) C_ B
10	6, 9	nfan 1618	. . 3 |- F/ x ( ( Fun F /\ A C_ dom F ) /\ ( F " A ) C_ B )
11		funfvima2 5951	. . . 4 |- ( ( Fun F /\ A C_ dom F ) -> ( x e. A -> ( F ` x ) e. ( F " A ) ) )
12		ssel 3242	. . . 4 |- ( ( F " A ) C_ B -> ( ( F ` x ) e. ( F " A ) -> ( F ` x ) e. B ) )
13	11, 12	sylan9 413	. . 3 |- ( ( ( Fun F /\ A C_ dom F ) /\ ( F " A ) C_ B ) -> ( x e. A -> ( F ` x ) e. B ) )
14	10, 13	ralrimi 2621	. 2 |- ( ( ( Fun F /\ A C_ dom F ) /\ ( F " A ) C_ B ) -> A. x e. A ( F ` x ) e. B )
15	3, 1	dfimafnf 5955	. . . 4 |- ( ( Fun F /\ A C_ dom F ) -> ( F " A ) = { y | E. x e. A y = ( F ` x ) } )
16	15	adantr 276	. . 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 6358	. . . 4 |- ( A. x e. A ( F ` x ) e. B -> { y | E. x e. A y = ( F ` x ) } C_ B )
18	17	adantl 277	. . 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 3284	. 2 |- ( ( ( Fun F /\ A C_ dom F ) /\ A. x e. A ( F ` x ) e. B ) -> ( F " A ) C_ B )
20	14, 19	impbida 604	1 |- ( ( Fun F /\ A C_ dom F ) -> ( ( F " A ) C_ B <-> A. x e. A ( F ` x ) e. B ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   {cab 2224   F/_wnfc 2379   A.wral 2528   E.wrex 2529   C_ wss 3220   dom cdm 4774   "cima 4777   Fun wfun 5371   ` cfv 5377

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-opab 4193  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-fv 5385

This theorem is used by:  ballotfilem7  13281