Theorem neissex 23475

Description: For any neighborhood 𝑁 of 𝑆, there is a neighborhood 𝑥 of 𝑆 such that 𝑁 is a neighborhood of all subsets of 𝑥. Generalization to subsets of Property V_iv of [BourbakiTop1] p. I.3. (Contributed by FL, 2-Oct-2006.)

Assertion

Ref	Expression
neissex	⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → ∃𝑥 ∈ ((nei‘𝐽)‘𝑆)∀𝑦(𝑦 ⊆ 𝑥 → 𝑁 ∈ ((nei‘𝐽)‘𝑦)))

Distinct variable groups:   𝑥,𝑦,𝐽   𝑥,𝑁,𝑦   𝑥,𝑆,𝑦

Proof of Theorem neissex

Step	Hyp	Ref	Expression
1		neii2 23456	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → ∃𝑥 ∈ 𝐽 (𝑆 ⊆ 𝑥 ∧ 𝑥 ⊆ 𝑁))
2		opnneiss 23466	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽 ∧ 𝑆 ⊆ 𝑥) → 𝑥 ∈ ((nei‘𝐽)‘𝑆))
3	2	3expb 1138	. . . 4 ⊢ ((𝐽 ∈ Top ∧ (𝑥 ∈ 𝐽 ∧ 𝑆 ⊆ 𝑥)) → 𝑥 ∈ ((nei‘𝐽)‘𝑆))
4	3	adantrrr 738	. . 3 ⊢ ((𝐽 ∈ Top ∧ (𝑥 ∈ 𝐽 ∧ (𝑆 ⊆ 𝑥 ∧ 𝑥 ⊆ 𝑁))) → 𝑥 ∈ ((nei‘𝐽)‘𝑆))
5	4	adantlr 728	. 2 ⊢ (((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) ∧ (𝑥 ∈ 𝐽 ∧ (𝑆 ⊆ 𝑥 ∧ 𝑥 ⊆ 𝑁))) → 𝑥 ∈ ((nei‘𝐽)‘𝑆))
6		simplll 787	. . . . . 6 ⊢ ((((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) ∧ (𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝑁)) ∧ 𝑦 ⊆ 𝑥) → 𝐽 ∈ Top)
7		simpll 779	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) ∧ 𝑥 ∈ 𝐽) → 𝐽 ∈ Top)
8		simpr 490	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) ∧ 𝑥 ∈ 𝐽) → 𝑥 ∈ 𝐽)
9		eqid 2763	. . . . . . . . . . . 12 ⊢ ∪ 𝐽 = ∪ 𝐽
10	9	neii1 23454	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → 𝑁 ⊆ ∪ 𝐽)
11	10	adantr 486	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) ∧ 𝑥 ∈ 𝐽) → 𝑁 ⊆ ∪ 𝐽)
12	9	opnssneib 23463	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽 ∧ 𝑁 ⊆ ∪ 𝐽) → (𝑥 ⊆ 𝑁 ↔ 𝑁 ∈ ((nei‘𝐽)‘𝑥)))
13	7, 8, 11, 12	syl3anc 1399	. . . . . . . . 9 ⊢ (((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) ∧ 𝑥 ∈ 𝐽) → (𝑥 ⊆ 𝑁 ↔ 𝑁 ∈ ((nei‘𝐽)‘𝑥)))
14	13	biimpa 482	. . . . . . . 8 ⊢ ((((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) ∧ 𝑥 ∈ 𝐽) ∧ 𝑥 ⊆ 𝑁) → 𝑁 ∈ ((nei‘𝐽)‘𝑥))
15	14	anasss 472	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) ∧ (𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝑁)) → 𝑁 ∈ ((nei‘𝐽)‘𝑥))
16	15	adantr 486	. . . . . 6 ⊢ ((((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) ∧ (𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝑁)) ∧ 𝑦 ⊆ 𝑥) → 𝑁 ∈ ((nei‘𝐽)‘𝑥))
17		simpr 490	. . . . . 6 ⊢ ((((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) ∧ (𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝑁)) ∧ 𝑦 ⊆ 𝑥) → 𝑦 ⊆ 𝑥)
18		neiss 23457	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑥) ∧ 𝑦 ⊆ 𝑥) → 𝑁 ∈ ((nei‘𝐽)‘𝑦))
19	6, 16, 17, 18	syl3anc 1399	. . . . 5 ⊢ ((((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) ∧ (𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝑁)) ∧ 𝑦 ⊆ 𝑥) → 𝑁 ∈ ((nei‘𝐽)‘𝑦))
20	19	ex 418	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) ∧ (𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝑁)) → (𝑦 ⊆ 𝑥 → 𝑁 ∈ ((nei‘𝐽)‘𝑦)))
21	20	adantrrl 737	. . 3 ⊢ (((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) ∧ (𝑥 ∈ 𝐽 ∧ (𝑆 ⊆ 𝑥 ∧ 𝑥 ⊆ 𝑁))) → (𝑦 ⊆ 𝑥 → 𝑁 ∈ ((nei‘𝐽)‘𝑦)))
22	21	alrimiv 1962	. 2 ⊢ (((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) ∧ (𝑥 ∈ 𝐽 ∧ (𝑆 ⊆ 𝑥 ∧ 𝑥 ⊆ 𝑁))) → ∀𝑦(𝑦 ⊆ 𝑥 → 𝑁 ∈ ((nei‘𝐽)‘𝑦)))
23	1, 5, 22	reximssdv 3184	1 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → ∃𝑥 ∈ ((nei‘𝐽)‘𝑆)∀𝑦(𝑦 ⊆ 𝑥 → 𝑁 ∈ ((nei‘𝐽)‘𝑦)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1568   ∈ wcel 2147  ∃wrex 3090   ⊆ wss 3900  ∪ cuni 4870  ‘cfv 6558  Topctop 23241  neicnei 23445

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-top 23242  df-nei 23446

This theorem is used by: (None)