Theorem fvelimab 5172

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 2560	. . . 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 2934	. . . . . . . 8 ⊢ ((B ⊆ A ∧ u ∈ B) → u ∈ A)
4		funfvex 5135	. . . . . . . . 9 ⊢ ((Fun 𝐹 ∧ u ∈ dom 𝐹) → (𝐹‘u) ∈ V)
5	4	funfni 4942	. . . . . . . 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 380	. . . . . 6 ⊢ (((𝐹 Fn A ∧ B ⊆ A) ∧ u ∈ B) → (𝐹‘u) ∈ V)
8		eleq1 2097	. . . . . 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 2427	. . . 4 ⊢ ((𝐹 Fn A ∧ B ⊆ A) → (∃u ∈ B (𝐹‘u) = 𝐶 → 𝐶 ∈ V))
11	10	imdistani 419	. . 3 ⊢ (((𝐹 Fn A ∧ B ⊆ A) ∧ ∃u ∈ B (𝐹‘u) = 𝐶) → ((𝐹 Fn A ∧ B ⊆ A) ∧ 𝐶 ∈ V))
12		eleq1 2097	. . . . . . 7 ⊢ (v = 𝐶 → (v ∈ (𝐹 “ B) ↔ 𝐶 ∈ (𝐹 “ B)))
13		eqeq2 2046	. . . . . . . 8 ⊢ (v = 𝐶 → ((𝐹‘u) = v ↔ (𝐹‘u) = 𝐶))
14	13	rexbidv 2321	. . . . . . 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 4939	. . . . . . . 8 ⊢ (𝐹 Fn A → Fun 𝐹)
18	17	adantr 261	. . . . . . 7 ⊢ ((𝐹 Fn A ∧ B ⊆ A) → Fun 𝐹)
19		fndm 4941	. . . . . . . . 9 ⊢ (𝐹 Fn A → dom 𝐹 = A)
20	19	sseq2d 2967	. . . . . . . 8 ⊢ (𝐹 Fn A → (B ⊆ dom 𝐹 ↔ B ⊆ A))
21	20	biimpar 281	. . . . . . 7 ⊢ ((𝐹 Fn A ∧ B ⊆ A) → B ⊆ dom 𝐹)
22		dfimafn 5165	. . . . . . 7 ⊢ ((Fun 𝐹 ∧ B ⊆ dom 𝐹) → (𝐹 “ B) = {v ∣ ∃u ∈ B (𝐹‘u) = v})
23	18, 21, 22	syl2anc 391	. . . . . 6 ⊢ ((𝐹 Fn A ∧ B ⊆ A) → (𝐹 “ B) = {v ∣ ∃u ∈ B (𝐹‘u) = v})
24	23	abeq2d 2147	. . . . 5 ⊢ ((𝐹 Fn A ∧ B ⊆ A) → (v ∈ (𝐹 “ B) ↔ ∃u ∈ B (𝐹‘u) = v))
25	16, 24	vtoclg 2607	. . . 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 824	. 2 ⊢ ((𝐹 Fn A ∧ B ⊆ A) → (𝐶 ∈ (𝐹 “ B) ↔ ∃u ∈ B (𝐹‘u) = 𝐶))
28		fveq2 5121	. . . 4 ⊢ (u = x → (𝐹‘u) = (𝐹‘x))
29	28	eqeq1d 2045	. . 3 ⊢ (u = x → ((𝐹‘u) = 𝐶 ↔ (𝐹‘x) = 𝐶))
30	29	cbvrexv 2528	. 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 1242   ∈ wcel 1390  {cab 2023  ∃wrex 2301  Vcvv 2551   ⊆ wss 2911  dom cdm 4288   “ cima 4291  Fun wfun 4839   Fn wfn 4840  ‘cfv 4845

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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-fv 4853

This theorem is referenced by:  ssimaex  5177  rexima  5337  ralima  5338  f1elima  5355  ovelimab  5593