Theorem neissex 14888

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 14872	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → ∃𝑥 ∈ 𝐽 (𝑆 ⊆ 𝑥 ∧ 𝑥 ⊆ 𝑁))
2		opnneiss 14881	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽 ∧ 𝑆 ⊆ 𝑥) → 𝑥 ∈ ((nei‘𝐽)‘𝑆))
3	2	3expb 1230	. . . 4 ⊢ ((𝐽 ∈ Top ∧ (𝑥 ∈ 𝐽 ∧ 𝑆 ⊆ 𝑥)) → 𝑥 ∈ ((nei‘𝐽)‘𝑆))
4	3	adantrrr 487	. . 3 ⊢ ((𝐽 ∈ Top ∧ (𝑥 ∈ 𝐽 ∧ (𝑆 ⊆ 𝑥 ∧ 𝑥 ⊆ 𝑁))) → 𝑥 ∈ ((nei‘𝐽)‘𝑆))
5	4	adantlr 477	. 2 ⊢ (((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) ∧ (𝑥 ∈ 𝐽 ∧ (𝑆 ⊆ 𝑥 ∧ 𝑥 ⊆ 𝑁))) → 𝑥 ∈ ((nei‘𝐽)‘𝑆))
6		simplll 535	. . . . . 6 ⊢ ((((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) ∧ (𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝑁)) ∧ 𝑦 ⊆ 𝑥) → 𝐽 ∈ Top)
7		simpll 527	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) ∧ 𝑥 ∈ 𝐽) → 𝐽 ∈ Top)
8		simpr 110	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) ∧ 𝑥 ∈ 𝐽) → 𝑥 ∈ 𝐽)
9		eqid 2231	. . . . . . . . . . . 12 ⊢ ∪ 𝐽 = ∪ 𝐽
10	9	neii1 14870	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → 𝑁 ⊆ ∪ 𝐽)
11	10	adantr 276	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) ∧ 𝑥 ∈ 𝐽) → 𝑁 ⊆ ∪ 𝐽)
12	9	opnssneib 14879	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽 ∧ 𝑁 ⊆ ∪ 𝐽) → (𝑥 ⊆ 𝑁 ↔ 𝑁 ∈ ((nei‘𝐽)‘𝑥)))
13	7, 8, 11, 12	syl3anc 1273	. . . . . . . . 9 ⊢ (((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) ∧ 𝑥 ∈ 𝐽) → (𝑥 ⊆ 𝑁 ↔ 𝑁 ∈ ((nei‘𝐽)‘𝑥)))
14	13	biimpa 296	. . . . . . . 8 ⊢ ((((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) ∧ 𝑥 ∈ 𝐽) ∧ 𝑥 ⊆ 𝑁) → 𝑁 ∈ ((nei‘𝐽)‘𝑥))
15	14	anasss 399	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) ∧ (𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝑁)) → 𝑁 ∈ ((nei‘𝐽)‘𝑥))
16	15	adantr 276	. . . . . 6 ⊢ ((((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) ∧ (𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝑁)) ∧ 𝑦 ⊆ 𝑥) → 𝑁 ∈ ((nei‘𝐽)‘𝑥))
17		simpr 110	. . . . . 6 ⊢ ((((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) ∧ (𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝑁)) ∧ 𝑦 ⊆ 𝑥) → 𝑦 ⊆ 𝑥)
18		neiss 14873	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑥) ∧ 𝑦 ⊆ 𝑥) → 𝑁 ∈ ((nei‘𝐽)‘𝑦))
19	6, 16, 17, 18	syl3anc 1273	. . . . 5 ⊢ ((((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) ∧ (𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝑁)) ∧ 𝑦 ⊆ 𝑥) → 𝑁 ∈ ((nei‘𝐽)‘𝑦))
20	19	ex 115	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) ∧ (𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝑁)) → (𝑦 ⊆ 𝑥 → 𝑁 ∈ ((nei‘𝐽)‘𝑦)))
21	20	adantrrl 486	. . 3 ⊢ (((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) ∧ (𝑥 ∈ 𝐽 ∧ (𝑆 ⊆ 𝑥 ∧ 𝑥 ⊆ 𝑁))) → (𝑦 ⊆ 𝑥 → 𝑁 ∈ ((nei‘𝐽)‘𝑦)))
22	21	alrimiv 1922	. 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 2636	1 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → ∃𝑥 ∈ ((nei‘𝐽)‘𝑆)∀𝑦(𝑦 ⊆ 𝑥 → 𝑁 ∈ ((nei‘𝐽)‘𝑦)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  ∀wal 1395   ∈ wcel 2202  ∃wrex 2511   ⊆ wss 3200  ∪ cuni 3893  ‘cfv 5326  Topctop 14720  neicnei 14861

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-top 14721  df-nei 14862

This theorem is referenced by: (None)