Theorem funimass4f 6332

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 5380	. . . . 5 |- F/ x Fun F
3		funimass4f.1	. . . . . 6 |- F/_ x A
4	1	nfdm 5006	. . . . . 6 |- F/_ x dom F
5	3, 4	nfss 3235	. . . . 5 |- F/ x A C_ dom F
6	2, 5	nfan 1614	. . . 4 |- F/ x ( Fun F /\ A C_ dom F )
7	1, 3	nfima 5114	. . . . 5 |- F/_ x ( F " A )
8		funimass4f.2	. . . . 5 |- F/_ x B
9	7, 8	nfss 3235	. . . 4 |- F/ x ( F " A ) C_ B
10	6, 9	nfan 1614	. . 3 |- F/ x ( ( Fun F /\ A C_ dom F ) /\ ( F " A ) C_ B )
11		funfvima2 5924	. . . 4 |- ( ( Fun F /\ A C_ dom F ) -> ( x e. A -> ( F ` x ) e. ( F " A ) ) )
12		ssel 3236	. . . 4 |- ( ( F " A ) C_ B -> ( ( F ` x ) e. ( F " A ) -> ( F ` x ) e. B ) )
13	11, 12	sylan9 409	. . 3 |- ( ( ( Fun F /\ A C_ dom F ) /\ ( F " A ) C_ B ) -> ( x e. A -> ( F ` x ) e. B ) )
14	10, 13	ralrimi 2615	. 2 |- ( ( ( Fun F /\ A C_ dom F ) /\ ( F " A ) C_ B ) -> A. x e. A ( F ` x ) e. B )
15	3, 1	dfimafnf 5928	. . . 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 6331	. . . 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 3278	. 2 |- ( ( ( Fun F /\ A C_ dom F ) /\ A. x e. A ( F ` x ) e. B ) -> ( F " A ) C_ B )
20	14, 19	impbida 600	1 |- ( ( Fun F /\ A C_ dom F ) -> ( ( F " A ) C_ B <-> A. x e. A ( F ` x ) e. B ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2205   {cab 2220   F/_wnfc 2373   A.wral 2522   E.wrex 2523   C_ wss 3214   dom cdm 4754   "cima 4757   Fun wfun 5351   ` cfv 5357

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-sbc 3046  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-opab 4177  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-fv 5365

This theorem is referenced by:  ballotfilem7  13223