Theorem fvelimab 5614

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 2771	. . . 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 3175	. . . . . . . 8 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝑢 ∈ 𝐵) → 𝑢 ∈ 𝐴)
4		funfvex 5572	. . . . . . . . 9 ⊢ ((Fun 𝐹 ∧ 𝑢 ∈ dom 𝐹) → (𝐹‘𝑢) ∈ V)
5	4	funfni 5355	. . . . . . . 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 400	. . . . . 6 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ 𝑢 ∈ 𝐵) → (𝐹‘𝑢) ∈ V)
8		eleq1 2256	. . . . . 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 2611	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (∃𝑢 ∈ 𝐵 (𝐹‘𝑢) = 𝐶 → 𝐶 ∈ V))
11	10	imdistani 445	. . 3 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ ∃𝑢 ∈ 𝐵 (𝐹‘𝑢) = 𝐶) → ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ 𝐶 ∈ V))
12		eleq1 2256	. . . . . . 7 ⊢ (𝑣 = 𝐶 → (𝑣 ∈ (𝐹 “ 𝐵) ↔ 𝐶 ∈ (𝐹 “ 𝐵)))
13		eqeq2 2203	. . . . . . . 8 ⊢ (𝑣 = 𝐶 → ((𝐹‘𝑢) = 𝑣 ↔ (𝐹‘𝑢) = 𝐶))
14	13	rexbidv 2495	. . . . . . 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 5352	. . . . . . . 8 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹)
18	17	adantr 276	. . . . . . 7 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → Fun 𝐹)
19		fndm 5354	. . . . . . . . 9 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
20	19	sseq2d 3210	. . . . . . . 8 ⊢ (𝐹 Fn 𝐴 → (𝐵 ⊆ dom 𝐹 ↔ 𝐵 ⊆ 𝐴))
21	20	biimpar 297	. . . . . . 7 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → 𝐵 ⊆ dom 𝐹)
22		dfimafn 5606	. . . . . . 7 ⊢ ((Fun 𝐹 ∧ 𝐵 ⊆ dom 𝐹) → (𝐹 “ 𝐵) = {𝑣 ∣ ∃𝑢 ∈ 𝐵 (𝐹‘𝑢) = 𝑣})
23	18, 21, 22	syl2anc 411	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝐹 “ 𝐵) = {𝑣 ∣ ∃𝑢 ∈ 𝐵 (𝐹‘𝑢) = 𝑣})
24	23	abeq2d 2306	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝑣 ∈ (𝐹 “ 𝐵) ↔ ∃𝑢 ∈ 𝐵 (𝐹‘𝑢) = 𝑣))
25	16, 24	vtoclg 2821	. . . 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 917	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝐶 ∈ (𝐹 “ 𝐵) ↔ ∃𝑢 ∈ 𝐵 (𝐹‘𝑢) = 𝐶))
28		fveq2 5555	. . . 4 ⊢ (𝑢 = 𝑥 → (𝐹‘𝑢) = (𝐹‘𝑥))
29	28	eqeq1d 2202	. . 3 ⊢ (𝑢 = 𝑥 → ((𝐹‘𝑢) = 𝐶 ↔ (𝐹‘𝑥) = 𝐶))
30	29	cbvrexv 2727	. 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))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1364   ∈ wcel 2164  {cab 2179  ∃wrex 2473  Vcvv 2760   ⊆ wss 3154  dom cdm 4660   “ cima 4663  Fun wfun 5249   Fn wfn 5250  ‘cfv 5255

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2987  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-br 4031  df-opab 4092  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-fv 5263

This theorem is referenced by:  ssimaex  5619  foima2  5795  rexima  5798  ralima  5799  f1elima  5817  ovelimab  6071