Theorem neissex 23021

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 23002	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → ∃𝑥 ∈ 𝐽 (𝑆 ⊆ 𝑥 ∧ 𝑥 ⊆ 𝑁))
2		opnneiss 23012	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽 ∧ 𝑆 ⊆ 𝑥) → 𝑥 ∈ ((nei‘𝐽)‘𝑆))
3	2	3expb 1120	. . . 4 ⊢ ((𝐽 ∈ Top ∧ (𝑥 ∈ 𝐽 ∧ 𝑆 ⊆ 𝑥)) → 𝑥 ∈ ((nei‘𝐽)‘𝑆))
4	3	adantrrr 725	. . 3 ⊢ ((𝐽 ∈ Top ∧ (𝑥 ∈ 𝐽 ∧ (𝑆 ⊆ 𝑥 ∧ 𝑥 ⊆ 𝑁))) → 𝑥 ∈ ((nei‘𝐽)‘𝑆))
5	4	adantlr 715	. 2 ⊢ (((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) ∧ (𝑥 ∈ 𝐽 ∧ (𝑆 ⊆ 𝑥 ∧ 𝑥 ⊆ 𝑁))) → 𝑥 ∈ ((nei‘𝐽)‘𝑆))
6		simplll 774	. . . . . 6 ⊢ ((((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) ∧ (𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝑁)) ∧ 𝑦 ⊆ 𝑥) → 𝐽 ∈ Top)
7		simpll 766	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) ∧ 𝑥 ∈ 𝐽) → 𝐽 ∈ Top)
8		simpr 484	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) ∧ 𝑥 ∈ 𝐽) → 𝑥 ∈ 𝐽)
9		eqid 2730	. . . . . . . . . . . 12 ⊢ ∪ 𝐽 = ∪ 𝐽
10	9	neii1 23000	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → 𝑁 ⊆ ∪ 𝐽)
11	10	adantr 480	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) ∧ 𝑥 ∈ 𝐽) → 𝑁 ⊆ ∪ 𝐽)
12	9	opnssneib 23009	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽 ∧ 𝑁 ⊆ ∪ 𝐽) → (𝑥 ⊆ 𝑁 ↔ 𝑁 ∈ ((nei‘𝐽)‘𝑥)))
13	7, 8, 11, 12	syl3anc 1373	. . . . . . . . 9 ⊢ (((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) ∧ 𝑥 ∈ 𝐽) → (𝑥 ⊆ 𝑁 ↔ 𝑁 ∈ ((nei‘𝐽)‘𝑥)))
14	13	biimpa 476	. . . . . . . 8 ⊢ ((((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) ∧ 𝑥 ∈ 𝐽) ∧ 𝑥 ⊆ 𝑁) → 𝑁 ∈ ((nei‘𝐽)‘𝑥))
15	14	anasss 466	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) ∧ (𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝑁)) → 𝑁 ∈ ((nei‘𝐽)‘𝑥))
16	15	adantr 480	. . . . . 6 ⊢ ((((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) ∧ (𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝑁)) ∧ 𝑦 ⊆ 𝑥) → 𝑁 ∈ ((nei‘𝐽)‘𝑥))
17		simpr 484	. . . . . 6 ⊢ ((((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) ∧ (𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝑁)) ∧ 𝑦 ⊆ 𝑥) → 𝑦 ⊆ 𝑥)
18		neiss 23003	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑥) ∧ 𝑦 ⊆ 𝑥) → 𝑁 ∈ ((nei‘𝐽)‘𝑦))
19	6, 16, 17, 18	syl3anc 1373	. . . . 5 ⊢ ((((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) ∧ (𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝑁)) ∧ 𝑦 ⊆ 𝑥) → 𝑁 ∈ ((nei‘𝐽)‘𝑦))
20	19	ex 412	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) ∧ (𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝑁)) → (𝑦 ⊆ 𝑥 → 𝑁 ∈ ((nei‘𝐽)‘𝑦)))
21	20	adantrrl 724	. . 3 ⊢ (((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) ∧ (𝑥 ∈ 𝐽 ∧ (𝑆 ⊆ 𝑥 ∧ 𝑥 ⊆ 𝑁))) → (𝑦 ⊆ 𝑥 → 𝑁 ∈ ((nei‘𝐽)‘𝑦)))
22	21	alrimiv 1927	. 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 3152	1 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → ∃𝑥 ∈ ((nei‘𝐽)‘𝑆)∀𝑦(𝑦 ⊆ 𝑥 → 𝑁 ∈ ((nei‘𝐽)‘𝑦)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1538   ∈ wcel 2109  ∃wrex 3054   ⊆ wss 3917  ∪ cuni 4874  ‘cfv 6514  Topctop 22787  neicnei 22991

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-top 22788  df-nei 22992

This theorem is referenced by: (None)