Theorem fvelimab 5485

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 2700	. . . 4 ⊢ (𝐶 ∈ (𝐹 “ 𝐵) → 𝐶 ∈ V)
2	1	anim2i 340	. . 3 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ 𝐶 ∈ (𝐹 “ 𝐵)) → ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ 𝐶 ∈ V))
3		ssel2 3097	. . . . . . . 8 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝑢 ∈ 𝐵) → 𝑢 ∈ 𝐴)
4		funfvex 5446	. . . . . . . . 9 ⊢ ((Fun 𝐹 ∧ 𝑢 ∈ dom 𝐹) → (𝐹‘𝑢) ∈ V)
5	4	funfni 5231	. . . . . . . 8 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑢 ∈ 𝐴) → (𝐹‘𝑢) ∈ V)
6	3, 5	sylan2 284	. . . . . . 7 ⊢ ((𝐹 Fn 𝐴 ∧ (𝐵 ⊆ 𝐴 ∧ 𝑢 ∈ 𝐵)) → (𝐹‘𝑢) ∈ V)
7	6	anassrs 398	. . . . . 6 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ 𝑢 ∈ 𝐵) → (𝐹‘𝑢) ∈ V)
8		eleq1 2203	. . . . . 6 ⊢ ((𝐹‘𝑢) = 𝐶 → ((𝐹‘𝑢) ∈ V ↔ 𝐶 ∈ V))
9	7, 8	syl5ibcom 154	. . . . 5 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ 𝑢 ∈ 𝐵) → ((𝐹‘𝑢) = 𝐶 → 𝐶 ∈ V))
10	9	rexlimdva 2552	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (∃𝑢 ∈ 𝐵 (𝐹‘𝑢) = 𝐶 → 𝐶 ∈ V))
11	10	imdistani 442	. . 3 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ ∃𝑢 ∈ 𝐵 (𝐹‘𝑢) = 𝐶) → ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ 𝐶 ∈ V))
12		eleq1 2203	. . . . . . 7 ⊢ (𝑣 = 𝐶 → (𝑣 ∈ (𝐹 “ 𝐵) ↔ 𝐶 ∈ (𝐹 “ 𝐵)))
13		eqeq2 2150	. . . . . . . 8 ⊢ (𝑣 = 𝐶 → ((𝐹‘𝑢) = 𝑣 ↔ (𝐹‘𝑢) = 𝐶))
14	13	rexbidv 2439	. . . . . . 7 ⊢ (𝑣 = 𝐶 → (∃𝑢 ∈ 𝐵 (𝐹‘𝑢) = 𝑣 ↔ ∃𝑢 ∈ 𝐵 (𝐹‘𝑢) = 𝐶))
15	12, 14	bibi12d 234	. . . . . 6 ⊢ (𝑣 = 𝐶 → ((𝑣 ∈ (𝐹 “ 𝐵) ↔ ∃𝑢 ∈ 𝐵 (𝐹‘𝑢) = 𝑣) ↔ (𝐶 ∈ (𝐹 “ 𝐵) ↔ ∃𝑢 ∈ 𝐵 (𝐹‘𝑢) = 𝐶)))
16	15	imbi2d 229	. . . . 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 5228	. . . . . . . 8 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹)
18	17	adantr 274	. . . . . . 7 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → Fun 𝐹)
19		fndm 5230	. . . . . . . . 9 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
20	19	sseq2d 3132	. . . . . . . 8 ⊢ (𝐹 Fn 𝐴 → (𝐵 ⊆ dom 𝐹 ↔ 𝐵 ⊆ 𝐴))
21	20	biimpar 295	. . . . . . 7 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → 𝐵 ⊆ dom 𝐹)
22		dfimafn 5478	. . . . . . 7 ⊢ ((Fun 𝐹 ∧ 𝐵 ⊆ dom 𝐹) → (𝐹 “ 𝐵) = {𝑣 ∣ ∃𝑢 ∈ 𝐵 (𝐹‘𝑢) = 𝑣})
23	18, 21, 22	syl2anc 409	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝐹 “ 𝐵) = {𝑣 ∣ ∃𝑢 ∈ 𝐵 (𝐹‘𝑢) = 𝑣})
24	23	abeq2d 2253	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝑣 ∈ (𝐹 “ 𝐵) ↔ ∃𝑢 ∈ 𝐵 (𝐹‘𝑢) = 𝑣))
25	16, 24	vtoclg 2749	. . . 4 ⊢ (𝐶 ∈ V → ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝐶 ∈ (𝐹 “ 𝐵) ↔ ∃𝑢 ∈ 𝐵 (𝐹‘𝑢) = 𝐶)))
26	25	impcom 124	. . 3 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ 𝐶 ∈ V) → (𝐶 ∈ (𝐹 “ 𝐵) ↔ ∃𝑢 ∈ 𝐵 (𝐹‘𝑢) = 𝐶))
27	2, 11, 26	pm5.21nd 902	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝐶 ∈ (𝐹 “ 𝐵) ↔ ∃𝑢 ∈ 𝐵 (𝐹‘𝑢) = 𝐶))
28		fveq2 5429	. . . 4 ⊢ (𝑢 = 𝑥 → (𝐹‘𝑢) = (𝐹‘𝑥))
29	28	eqeq1d 2149	. . 3 ⊢ (𝑢 = 𝑥 → ((𝐹‘𝑢) = 𝐶 ↔ (𝐹‘𝑥) = 𝐶))
30	29	cbvrexv 2658	. 2 ⊢ (∃𝑢 ∈ 𝐵 (𝐹‘𝑢) = 𝐶 ↔ ∃𝑥 ∈ 𝐵 (𝐹‘𝑥) = 𝐶)
31	27, 30	syl6bb 195	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝐶 ∈ (𝐹 “ 𝐵) ↔ ∃𝑥 ∈ 𝐵 (𝐹‘𝑥) = 𝐶))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1332   ∈ wcel 1481  {cab 2126  ∃wrex 2418  Vcvv 2689   ⊆ wss 3076  dom cdm 4547   “ cima 4550  Fun wfun 5125   Fn wfn 5126  ‘cfv 5131

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-sbc 2914  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-br 3938  df-opab 3998  df-id 4223  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-fv 5139

This theorem is referenced by:  ssimaex  5490  foima2  5661  rexima  5664  ralima  5665  f1elima  5682  ovelimab  5929