Theorem opnssneib 23123

Description: Any superset of an open set is a neighborhood of it. (Contributed by NM, 14-Feb-2007.)

Hypothesis

Ref	Expression
neips.1	⊢ 𝑋 = ∪ 𝐽

Assertion

Ref	Expression
opnssneib	⊢ ((𝐽 ∈ Top ∧ 𝑆 ∈ 𝐽 ∧ 𝑁 ⊆ 𝑋) → (𝑆 ⊆ 𝑁 ↔ 𝑁 ∈ ((nei‘𝐽)‘𝑆)))

Proof of Theorem opnssneib

Dummy variable 𝑔 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simplr 769	. . . . . 6 ⊢ (((𝑆 ∈ 𝐽 ∧ 𝑁 ⊆ 𝑋) ∧ 𝑆 ⊆ 𝑁) → 𝑁 ⊆ 𝑋)
2		sseq2 4010	. . . . . . . . . 10 ⊢ (𝑔 = 𝑆 → (𝑆 ⊆ 𝑔 ↔ 𝑆 ⊆ 𝑆))
3		sseq1 4009	. . . . . . . . . 10 ⊢ (𝑔 = 𝑆 → (𝑔 ⊆ 𝑁 ↔ 𝑆 ⊆ 𝑁))
4	2, 3	anbi12d 632	. . . . . . . . 9 ⊢ (𝑔 = 𝑆 → ((𝑆 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑁) ↔ (𝑆 ⊆ 𝑆 ∧ 𝑆 ⊆ 𝑁)))
5		ssid 4006	. . . . . . . . . 10 ⊢ 𝑆 ⊆ 𝑆
6	5	biantrur 530	. . . . . . . . 9 ⊢ (𝑆 ⊆ 𝑁 ↔ (𝑆 ⊆ 𝑆 ∧ 𝑆 ⊆ 𝑁))
7	4, 6	bitr4di 289	. . . . . . . 8 ⊢ (𝑔 = 𝑆 → ((𝑆 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑁) ↔ 𝑆 ⊆ 𝑁))
8	7	rspcev 3622	. . . . . . 7 ⊢ ((𝑆 ∈ 𝐽 ∧ 𝑆 ⊆ 𝑁) → ∃𝑔 ∈ 𝐽 (𝑆 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑁))
9	8	adantlr 715	. . . . . 6 ⊢ (((𝑆 ∈ 𝐽 ∧ 𝑁 ⊆ 𝑋) ∧ 𝑆 ⊆ 𝑁) → ∃𝑔 ∈ 𝐽 (𝑆 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑁))
10	1, 9	jca 511	. . . . 5 ⊢ (((𝑆 ∈ 𝐽 ∧ 𝑁 ⊆ 𝑋) ∧ 𝑆 ⊆ 𝑁) → (𝑁 ⊆ 𝑋 ∧ ∃𝑔 ∈ 𝐽 (𝑆 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑁)))
11	10	ex 412	. . . 4 ⊢ ((𝑆 ∈ 𝐽 ∧ 𝑁 ⊆ 𝑋) → (𝑆 ⊆ 𝑁 → (𝑁 ⊆ 𝑋 ∧ ∃𝑔 ∈ 𝐽 (𝑆 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑁))))
12	11	3adant1 1131	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ∈ 𝐽 ∧ 𝑁 ⊆ 𝑋) → (𝑆 ⊆ 𝑁 → (𝑁 ⊆ 𝑋 ∧ ∃𝑔 ∈ 𝐽 (𝑆 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑁))))
13		neips.1	. . . . . 6 ⊢ 𝑋 = ∪ 𝐽
14	13	eltopss 22913	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ∈ 𝐽) → 𝑆 ⊆ 𝑋)
15	13	isnei 23111	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑁 ∈ ((nei‘𝐽)‘𝑆) ↔ (𝑁 ⊆ 𝑋 ∧ ∃𝑔 ∈ 𝐽 (𝑆 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑁))))
16	14, 15	syldan 591	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ∈ 𝐽) → (𝑁 ∈ ((nei‘𝐽)‘𝑆) ↔ (𝑁 ⊆ 𝑋 ∧ ∃𝑔 ∈ 𝐽 (𝑆 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑁))))
17	16	3adant3 1133	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ∈ 𝐽 ∧ 𝑁 ⊆ 𝑋) → (𝑁 ∈ ((nei‘𝐽)‘𝑆) ↔ (𝑁 ⊆ 𝑋 ∧ ∃𝑔 ∈ 𝐽 (𝑆 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑁))))
18	12, 17	sylibrd 259	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ∈ 𝐽 ∧ 𝑁 ⊆ 𝑋) → (𝑆 ⊆ 𝑁 → 𝑁 ∈ ((nei‘𝐽)‘𝑆)))
19		ssnei 23118	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → 𝑆 ⊆ 𝑁)
20	19	ex 412	. . 3 ⊢ (𝐽 ∈ Top → (𝑁 ∈ ((nei‘𝐽)‘𝑆) → 𝑆 ⊆ 𝑁))
21	20	3ad2ant1 1134	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ∈ 𝐽 ∧ 𝑁 ⊆ 𝑋) → (𝑁 ∈ ((nei‘𝐽)‘𝑆) → 𝑆 ⊆ 𝑁))
22	18, 21	impbid 212	1 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ∈ 𝐽 ∧ 𝑁 ⊆ 𝑋) → (𝑆 ⊆ 𝑁 ↔ 𝑁 ∈ ((nei‘𝐽)‘𝑆)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1540   ∈ wcel 2108  ∃wrex 3070   ⊆ wss 3951  ∪ cuni 4907  ‘cfv 6561  Topctop 22899  neicnei 23105

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-top 22900  df-nei 23106

This theorem is referenced by:  neissex  23135