Theorem fvelimab 5759

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 2833	. . . 4 ⊢ (𝐶 ∈ (𝐹 “ 𝐵) → 𝐶 ∈ V)
2	1	anim2i 342	. . 3 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ 𝐶 ∈ (𝐹 “ 𝐵)) → ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ 𝐶 ∈ V))
3		ssel2 3243	. . . . . . . 8 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝑢 ∈ 𝐵) → 𝑢 ∈ 𝐴)
4		funfvex 5712	. . . . . . . . 9 ⊢ ((Fun 𝐹 ∧ 𝑢 ∈ dom 𝐹) → (𝐹‘𝑢) ∈ V)
5	4	funfni 5483	. . . . . . . 8 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑢 ∈ 𝐴) → (𝐹‘𝑢) ∈ V)
6	3, 5	sylan2 286	. . . . . . 7 ⊢ ((𝐹 Fn 𝐴 ∧ (𝐵 ⊆ 𝐴 ∧ 𝑢 ∈ 𝐵)) → (𝐹‘𝑢) ∈ V)
7	6	anassrs 404	. . . . . 6 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ 𝑢 ∈ 𝐵) → (𝐹‘𝑢) ∈ V)
8		eleq1 2301	. . . . . 6 ⊢ ((𝐹‘𝑢) = 𝐶 → ((𝐹‘𝑢) ∈ V ↔ 𝐶 ∈ V))
9	7, 8	syl5ibcom 155	. . . . 5 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ 𝑢 ∈ 𝐵) → ((𝐹‘𝑢) = 𝐶 → 𝐶 ∈ V))
10	9	rexlimdva 2668	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (∃𝑢 ∈ 𝐵 (𝐹‘𝑢) = 𝐶 → 𝐶 ∈ V))
11	10	imdistani 449	. . 3 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ ∃𝑢 ∈ 𝐵 (𝐹‘𝑢) = 𝐶) → ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ 𝐶 ∈ V))
12		eleq1 2301	. . . . . . 7 ⊢ (𝑣 = 𝐶 → (𝑣 ∈ (𝐹 “ 𝐵) ↔ 𝐶 ∈ (𝐹 “ 𝐵)))
13		eqeq2 2248	. . . . . . . 8 ⊢ (𝑣 = 𝐶 → ((𝐹‘𝑢) = 𝑣 ↔ (𝐹‘𝑢) = 𝐶))
14	13	rexbidv 2551	. . . . . . 7 ⊢ (𝑣 = 𝐶 → (∃𝑢 ∈ 𝐵 (𝐹‘𝑢) = 𝑣 ↔ ∃𝑢 ∈ 𝐵 (𝐹‘𝑢) = 𝐶))
15	12, 14	bibi12d 235	. . . . . 6 ⊢ (𝑣 = 𝐶 → ((𝑣 ∈ (𝐹 “ 𝐵) ↔ ∃𝑢 ∈ 𝐵 (𝐹‘𝑢) = 𝑣) ↔ (𝐶 ∈ (𝐹 “ 𝐵) ↔ ∃𝑢 ∈ 𝐵 (𝐹‘𝑢) = 𝐶)))
16	15	imbi2d 230	. . . . 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 5478	. . . . . . . 8 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹)
18	17	adantr 276	. . . . . . 7 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → Fun 𝐹)
19		fndm 5480	. . . . . . . . 9 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
20	19	sseq2d 3278	. . . . . . . 8 ⊢ (𝐹 Fn 𝐴 → (𝐵 ⊆ dom 𝐹 ↔ 𝐵 ⊆ 𝐴))
21	20	biimpar 297	. . . . . . 7 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → 𝐵 ⊆ dom 𝐹)
22		dfimafn 5751	. . . . . . 7 ⊢ ((Fun 𝐹 ∧ 𝐵 ⊆ dom 𝐹) → (𝐹 “ 𝐵) = {𝑣 ∣ ∃𝑢 ∈ 𝐵 (𝐹‘𝑢) = 𝑣})
23	18, 21, 22	syl2anc 415	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝐹 “ 𝐵) = {𝑣 ∣ ∃𝑢 ∈ 𝐵 (𝐹‘𝑢) = 𝑣})
24	23	abeq2d 2351	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝑣 ∈ (𝐹 “ 𝐵) ↔ ∃𝑢 ∈ 𝐵 (𝐹‘𝑢) = 𝑣))
25	16, 24	vtoclg 2883	. . . 4 ⊢ (𝐶 ∈ V → ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝐶 ∈ (𝐹 “ 𝐵) ↔ ∃𝑢 ∈ 𝐵 (𝐹‘𝑢) = 𝐶)))
26	25	impcom 125	. . 3 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ 𝐶 ∈ V) → (𝐶 ∈ (𝐹 “ 𝐵) ↔ ∃𝑢 ∈ 𝐵 (𝐹‘𝑢) = 𝐶))
27	2, 11, 26	pm5.21nd 928	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝐶 ∈ (𝐹 “ 𝐵) ↔ ∃𝑢 ∈ 𝐵 (𝐹‘𝑢) = 𝐶))
28		fveq2 5695	. . . 4 ⊢ (𝑢 = 𝑥 → (𝐹‘𝑢) = (𝐹‘𝑥))
29	28	eqeq1d 2247	. . 3 ⊢ (𝑢 = 𝑥 → ((𝐹‘𝑢) = 𝐶 ↔ (𝐹‘𝑥) = 𝐶))
30	29	cbvrexv 2787	. 2 ⊢ (∃𝑢 ∈ 𝐵 (𝐹‘𝑢) = 𝐶 ↔ ∃𝑥 ∈ 𝐵 (𝐹‘𝑥) = 𝐶)
31	27, 30	bitrdi 196	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝐶 ∈ (𝐹 “ 𝐵) ↔ ∃𝑥 ∈ 𝐵 (𝐹‘𝑥) = 𝐶))

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1402   ∈ wcel 2209  {cab 2224  ∃wrex 2529  Vcvv 2821   ⊆ wss 3220  dom cdm 4774   “ cima 4777  Fun wfun 5371   Fn wfn 5372  ‘cfv 5377

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-opab 4193  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-fv 5385

This theorem is used by:  ssimaex  5764  foima2  5957  rexima  5960  ralima  5961  f1elima  5979  ovelimab  6240  ballotfilemsima  13311