Theorem fvelimab 5702

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 2814	. . . 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 3222	. . . . . . . 8 |- ( ( B C_ A /\ u e. B ) -> u e. A )
4		funfvex 5656	. . . . . . . . 9 |- ( ( Fun F /\ u e. dom F ) -> ( F ` u ) e. _V )
5	4	funfni 5432	. . . . . . . 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 2294	. . . . . 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 2650	. . . 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 2294	. . . . . . 7 |- ( v = C -> ( v e. ( F " B ) <-> C e. ( F " B ) ) )
13		eqeq2 2241	. . . . . . . 8 |- ( v = C -> ( ( F ` u ) = v <-> ( F ` u ) = C ) )
14	13	rexbidv 2533	. . . . . . 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 5427	. . . . . . . 8 |- ( F Fn A -> Fun F )
18	17	adantr 276	. . . . . . 7 |- ( ( F Fn A /\ B C_ A ) -> Fun F )
19		fndm 5429	. . . . . . . . 9 |- ( F Fn A -> dom F = A )
20	19	sseq2d 3257	. . . . . . . 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 5694	. . . . . . 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 2344	. . . . 5 |- ( ( F Fn A /\ B C_ A ) -> ( v e. ( F " B ) <-> E. u e. B ( F ` u ) = v ) )
25	16, 24	vtoclg 2864	. . . 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 923	. 2 |- ( ( F Fn A /\ B C_ A ) -> ( C e. ( F " B ) <-> E. u e. B ( F ` u ) = C ) )
28		fveq2 5639	. . . 4 |- ( u = x -> ( F ` u ) = ( F ` x ) )
29	28	eqeq1d 2240	. . 3 |- ( u = x -> ( ( F ` u ) = C <-> ( F ` x ) = C ) )
30	29	cbvrexv 2768	. 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 1397   e. wcel 2202   {cab 2217   E.wrex 2511   _Vcvv 2802   C_ wss 3200   dom cdm 4725   "cima 4728   Fun wfun 5320   Fn wfn 5321   ` cfv 5326

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-sbc 3032  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-fv 5334

This theorem is referenced by:  ssimaex  5707  foima2  5891  rexima  5894  ralima  5895  f1elima  5913  ovelimab  6172