Theorem neissex 14401

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 14385	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → ∃𝑥 ∈ 𝐽 (𝑆 ⊆ 𝑥 ∧ 𝑥 ⊆ 𝑁))
2		opnneiss 14394	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽 ∧ 𝑆 ⊆ 𝑥) → 𝑥 ∈ ((nei‘𝐽)‘𝑆))
3	2	3expb 1206	. . . 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 2196	. . . . . . . . . . . 12 ⊢ ∪ 𝐽 = ∪ 𝐽
10	9	neii1 14383	. . . . . . . . . . 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 14392	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽 ∧ 𝑁 ⊆ ∪ 𝐽) → (𝑥 ⊆ 𝑁 ↔ 𝑁 ∈ ((nei‘𝐽)‘𝑥)))
13	7, 8, 11, 12	syl3anc 1249	. . . . . . . . 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 14386	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑥) ∧ 𝑦 ⊆ 𝑥) → 𝑁 ∈ ((nei‘𝐽)‘𝑦))
19	6, 16, 17, 18	syl3anc 1249	. . . . 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 1888	. 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 2601	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 1362   ∈ wcel 2167  ∃wrex 2476   ⊆ wss 3157  ∪ cuni 3839  ‘cfv 5258  Topctop 14233  neicnei 14374

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-pow 4207  ax-pr 4242

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-top 14234  df-nei 14375

This theorem is referenced by: (None)