Theorem neissex 13298

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 13282	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → ∃𝑥 ∈ 𝐽 (𝑆 ⊆ 𝑥 ∧ 𝑥 ⊆ 𝑁))
2		opnneiss 13291	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽 ∧ 𝑆 ⊆ 𝑥) → 𝑥 ∈ ((nei‘𝐽)‘𝑆))
3	2	3expb 1204	. . . 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 533	. . . . . 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 2177	. . . . . . . . . . . 12 ⊢ ∪ 𝐽 = ∪ 𝐽
10	9	neii1 13280	. . . . . . . . . . 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 13289	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽 ∧ 𝑁 ⊆ ∪ 𝐽) → (𝑥 ⊆ 𝑁 ↔ 𝑁 ∈ ((nei‘𝐽)‘𝑥)))
13	7, 8, 11, 12	syl3anc 1238	. . . . . . . . 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 13283	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑥) ∧ 𝑦 ⊆ 𝑥) → 𝑁 ∈ ((nei‘𝐽)‘𝑦))
19	6, 16, 17, 18	syl3anc 1238	. . . . 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 1874	. 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 2581	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 1351   ∈ wcel 2148  ∃wrex 2456   ⊆ wss 3129  ∪ cuni 3807  ‘cfv 5211  Topctop 13128  neicnei 13271

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-pow 4171  ax-pr 4205

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4289  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-rn 4633  df-res 4634  df-ima 4635  df-iota 5173  df-fun 5213  df-fn 5214  df-f 5215  df-f1 5216  df-fo 5217  df-f1o 5218  df-fv 5219  df-top 13129  df-nei 13272

This theorem is referenced by: (None)