Theorem fvelimab 5154

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 ⊆ A) → (𝐶 ∈ (𝐹 “ B) ↔ ∃x ∈ 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 2543	. . . 4 ⊢ (𝐶 ∈ (𝐹 “ B) → 𝐶 ∈ V)
2	1	anim2i 324	. . 3 ⊢ (((𝐹 Fn A ∧ B ⊆ A) ∧ 𝐶 ∈ (𝐹 “ B)) → ((𝐹 Fn A ∧ B ⊆ A) ∧ 𝐶 ∈ V))
3		ssel2 2917	. . . . . . . 8 ⊢ ((B ⊆ A ∧ u ∈ B) → u ∈ A)
4		funfvex 5117	. . . . . . . . 9 ⊢ ((Fun 𝐹 ∧ u ∈ dom 𝐹) → (𝐹‘u) ∈ V)
5	4	funfni 4925	. . . . . . . 8 ⊢ ((𝐹 Fn A ∧ u ∈ A) → (𝐹‘u) ∈ V)
6	3, 5	sylan2 270	. . . . . . 7 ⊢ ((𝐹 Fn A ∧ (B ⊆ A ∧ u ∈ B)) → (𝐹‘u) ∈ V)
7	6	anassrs 382	. . . . . 6 ⊢ (((𝐹 Fn A ∧ B ⊆ A) ∧ u ∈ B) → (𝐹‘u) ∈ V)
8		eleq1 2082	. . . . . 6 ⊢ ((𝐹‘u) = 𝐶 → ((𝐹‘u) ∈ V ↔ 𝐶 ∈ V))
9	7, 8	syl5ibcom 144	. . . . 5 ⊢ (((𝐹 Fn A ∧ B ⊆ A) ∧ u ∈ B) → ((𝐹‘u) = 𝐶 → 𝐶 ∈ V))
10	9	rexlimdva 2411	. . . 4 ⊢ ((𝐹 Fn A ∧ B ⊆ A) → (∃u ∈ B (𝐹‘u) = 𝐶 → 𝐶 ∈ V))
11	10	imdistani 422	. . 3 ⊢ (((𝐹 Fn A ∧ B ⊆ A) ∧ ∃u ∈ B (𝐹‘u) = 𝐶) → ((𝐹 Fn A ∧ B ⊆ A) ∧ 𝐶 ∈ V))
12		eleq1 2082	. . . . . . 7 ⊢ (v = 𝐶 → (v ∈ (𝐹 “ B) ↔ 𝐶 ∈ (𝐹 “ B)))
13		eqeq2 2031	. . . . . . . 8 ⊢ (v = 𝐶 → ((𝐹‘u) = v ↔ (𝐹‘u) = 𝐶))
14	13	rexbidv 2305	. . . . . . 7 ⊢ (v = 𝐶 → (∃u ∈ B (𝐹‘u) = v ↔ ∃u ∈ B (𝐹‘u) = 𝐶))
15	12, 14	bibi12d 224	. . . . . 6 ⊢ (v = 𝐶 → ((v ∈ (𝐹 “ B) ↔ ∃u ∈ B (𝐹‘u) = v) ↔ (𝐶 ∈ (𝐹 “ B) ↔ ∃u ∈ B (𝐹‘u) = 𝐶)))
16	15	imbi2d 219	. . . . 5 ⊢ (v = 𝐶 → (((𝐹 Fn A ∧ B ⊆ A) → (v ∈ (𝐹 “ B) ↔ ∃u ∈ B (𝐹‘u) = v)) ↔ ((𝐹 Fn A ∧ B ⊆ A) → (𝐶 ∈ (𝐹 “ B) ↔ ∃u ∈ B (𝐹‘u) = 𝐶))))
17		fnfun 4922	. . . . . . . 8 ⊢ (𝐹 Fn A → Fun 𝐹)
18	17	adantr 261	. . . . . . 7 ⊢ ((𝐹 Fn A ∧ B ⊆ A) → Fun 𝐹)
19		fndm 4924	. . . . . . . . 9 ⊢ (𝐹 Fn A → dom 𝐹 = A)
20	19	sseq2d 2950	. . . . . . . 8 ⊢ (𝐹 Fn A → (B ⊆ dom 𝐹 ↔ B ⊆ A))
21	20	biimpar 281	. . . . . . 7 ⊢ ((𝐹 Fn A ∧ B ⊆ A) → B ⊆ dom 𝐹)
22		dfimafn 5147	. . . . . . 7 ⊢ ((Fun 𝐹 ∧ B ⊆ dom 𝐹) → (𝐹 “ B) = {v ∣ ∃u ∈ B (𝐹‘u) = v})
23	18, 21, 22	syl2anc 393	. . . . . 6 ⊢ ((𝐹 Fn A ∧ B ⊆ A) → (𝐹 “ B) = {v ∣ ∃u ∈ B (𝐹‘u) = v})
24	23	abeq2d 2132	. . . . 5 ⊢ ((𝐹 Fn A ∧ B ⊆ A) → (v ∈ (𝐹 “ B) ↔ ∃u ∈ B (𝐹‘u) = v))
25	16, 24	vtoclg 2590	. . . 4 ⊢ (𝐶 ∈ V → ((𝐹 Fn A ∧ B ⊆ A) → (𝐶 ∈ (𝐹 “ B) ↔ ∃u ∈ B (𝐹‘u) = 𝐶)))
26	25	impcom 116	. . 3 ⊢ (((𝐹 Fn A ∧ B ⊆ A) ∧ 𝐶 ∈ V) → (𝐶 ∈ (𝐹 “ B) ↔ ∃u ∈ B (𝐹‘u) = 𝐶))
27	2, 11, 26	pm5.21nd 815	. 2 ⊢ ((𝐹 Fn A ∧ B ⊆ A) → (𝐶 ∈ (𝐹 “ B) ↔ ∃u ∈ B (𝐹‘u) = 𝐶))
28		fveq2 5103	. . . 4 ⊢ (u = x → (𝐹‘u) = (𝐹‘x))
29	28	eqeq1d 2030	. . 3 ⊢ (u = x → ((𝐹‘u) = 𝐶 ↔ (𝐹‘x) = 𝐶))
30	29	cbvrexv 2512	. 2 ⊢ (∃u ∈ B (𝐹‘u) = 𝐶 ↔ ∃x ∈ B (𝐹‘x) = 𝐶)
31	27, 30	syl6bb 185	1 ⊢ ((𝐹 Fn A ∧ B ⊆ A) → (𝐶 ∈ (𝐹 “ B) ↔ ∃x ∈ B (𝐹‘x) = 𝐶))

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1228   ∈ wcel 1374  {cab 2008  ∃wrex 2285  Vcvv 2535   ⊆ wss 2894  dom cdm 4272   “ cima 4275  Fun wfun 4823   Fn wfn 4824  ‘cfv 4829

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918

This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-sbc 2742  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-uni 3555  df-br 3739  df-opab 3793  df-id 4004  df-xp 4278  df-rel 4279  df-cnv 4280  df-co 4281  df-dm 4282  df-rn 4283  df-res 4284  df-ima 4285  df-iota 4794  df-fun 4831  df-fn 4832  df-fv 4837

This theorem is referenced by:  ssimaex  5159  rexima  5319  ralima  5320  f1elima  5337  ovelimab  5574