Theorem fvelimab 5658

Description: Function value in an image. (Contributed by NM, 20-Jan-2007.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) (Revised by David Abernethy, 17-Dec-2011.)

Assertion

Ref	Expression
fvelimab	|- ( ( F Fn A /\ B C_ A ) -> ( C e. ( F " B ) <-> E. x e. B ( F ` x ) = C ) )

Distinct variable groups:   x, B   x, C   x, F

Allowed substitution hint:   A( x)

Proof of Theorem fvelimab

Dummy variables v u are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 2788	. . . 4 |- ( C e. ( F " B ) -> C e. _V )
2	1	anim2i 342	. . 3 |- ( ( ( F Fn A /\ B C_ A ) /\ C e. ( F " B ) ) -> ( ( F Fn A /\ B C_ A ) /\ C e. _V ) )
3		ssel2 3196	. . . . . . . 8 |- ( ( B C_ A /\ u e. B ) -> u e. A )
4		funfvex 5616	. . . . . . . . 9 |- ( ( Fun F /\ u e. dom F ) -> ( F ` u ) e. _V )
5	4	funfni 5395	. . . . . . . 8 |- ( ( F Fn A /\ u e. A ) -> ( F ` u ) e. _V )
6	3, 5	sylan2 286	. . . . . . 7 |- ( ( F Fn A /\ ( B C_ A /\ u e. B ) ) -> ( F ` u ) e. _V )
7	6	anassrs 400	. . . . . 6 |- ( ( ( F Fn A /\ B C_ A ) /\ u e. B ) -> ( F ` u ) e. _V )
8		eleq1 2270	. . . . . 6 |- ( ( F ` u ) = C -> ( ( F ` u ) e. _V <-> C e. _V ) )
9	7, 8	syl5ibcom 155	. . . . 5 |- ( ( ( F Fn A /\ B C_ A ) /\ u e. B ) -> ( ( F ` u ) = C -> C e. _V ) )
10	9	rexlimdva 2625	. . . 4 |- ( ( F Fn A /\ B C_ A ) -> ( E. u e. B ( F ` u ) = C -> C e. _V ) )
11	10	imdistani 445	. . 3 |- ( ( ( F Fn A /\ B C_ A ) /\ E. u e. B ( F ` u ) = C ) -> ( ( F Fn A /\ B C_ A ) /\ C e. _V ) )
12		eleq1 2270	. . . . . . 7 |- ( v = C -> ( v e. ( F " B ) <-> C e. ( F " B ) ) )
13		eqeq2 2217	. . . . . . . 8 |- ( v = C -> ( ( F ` u ) = v <-> ( F ` u ) = C ) )
14	13	rexbidv 2509	. . . . . . 7 |- ( v = C -> ( E. u e. B ( F ` u ) = v <-> E. u e. B ( F ` u ) = C ) )
15	12, 14	bibi12d 235	. . . . . 6 |- ( v = C -> ( ( v e. ( F " B ) <-> E. u e. B ( F ` u ) = v ) <-> ( C e. ( F " B ) <-> E. u e. B ( F ` u ) = C ) ) )
16	15	imbi2d 230	. . . . 5 |- ( v = C -> ( ( ( F Fn A /\ B C_ A ) -> ( v e. ( F " B ) <-> E. u e. B ( F ` u ) = v ) ) <-> ( ( F Fn A /\ B C_ A ) -> ( C e. ( F " B ) <-> E. u e. B ( F ` u ) = C ) ) ) )
17		fnfun 5390	. . . . . . . 8 |- ( F Fn A -> Fun F )
18	17	adantr 276	. . . . . . 7 |- ( ( F Fn A /\ B C_ A ) -> Fun F )
19		fndm 5392	. . . . . . . . 9 |- ( F Fn A -> dom F = A )
20	19	sseq2d 3231	. . . . . . . 8 |- ( F Fn A -> ( B C_ dom F <-> B C_ A ) )
21	20	biimpar 297	. . . . . . 7 |- ( ( F Fn A /\ B C_ A ) -> B C_ dom F )
22		dfimafn 5650	. . . . . . 7 |- ( ( Fun F /\ B C_ dom F ) -> ( F " B ) = { v | E. u e. B ( F ` u ) = v } )
23	18, 21, 22	syl2anc 411	. . . . . 6 |- ( ( F Fn A /\ B C_ A ) -> ( F " B ) = { v | E. u e. B ( F ` u ) = v } )
24	23	abeq2d 2320	. . . . 5 |- ( ( F Fn A /\ B C_ A ) -> ( v e. ( F " B ) <-> E. u e. B ( F ` u ) = v ) )
25	16, 24	vtoclg 2838	. . . 4 |- ( C e. _V -> ( ( F Fn A /\ B C_ A ) -> ( C e. ( F " B ) <-> E. u e. B ( F ` u ) = C ) ) )
26	25	impcom 125	. . 3 |- ( ( ( F Fn A /\ B C_ A ) /\ C e. _V ) -> ( C e. ( F " B ) <-> E. u e. B ( F ` u ) = C ) )
27	2, 11, 26	pm5.21nd 918	. 2 |- ( ( F Fn A /\ B C_ A ) -> ( C e. ( F " B ) <-> E. u e. B ( F ` u ) = C ) )
28		fveq2 5599	. . . 4 |- ( u = x -> ( F ` u ) = ( F ` x ) )
29	28	eqeq1d 2216	. . 3 |- ( u = x -> ( ( F ` u ) = C <-> ( F ` x ) = C ) )
30	29	cbvrexv 2743	. 2 |- ( E. u e. B ( F ` u ) = C <-> E. x e. B ( F ` x ) = C )
31	27, 30	bitrdi 196	1 |- ( ( F Fn A /\ B C_ A ) -> ( C e. ( F " B ) <-> E. x e. B ( F ` x ) = C ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1373   e. wcel 2178   {cab 2193   E.wrex 2487   _Vcvv 2776   C_ wss 3174   dom cdm 4693   "cima 4696   Fun wfun 5284   Fn wfn 5285   ` cfv 5290

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-sbc 3006  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-br 4060  df-opab 4122  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-fv 5298

This theorem is referenced by:  ssimaex  5663  foima2  5843  rexima  5846  ralima  5847  f1elima  5865  ovelimab  6120