Theorem regsep 21936

Description: In a regular space, every neighborhood of a point contains a closed subneighborhood. (Contributed by Mario Carneiro, 25-Aug-2015.)

Assertion

Ref	Expression
regsep	⊢ ((𝐽 ∈ Reg ∧ 𝑈 ∈ 𝐽 ∧ 𝐴 ∈ 𝑈) → ∃𝑥 ∈ 𝐽 (𝐴 ∈ 𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝑈))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐽   𝑥,𝑈

Proof of Theorem regsep

Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		isreg 21934	. . . 4 ⊢ (𝐽 ∈ Reg ↔ (𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝐽 ∀𝑧 ∈ 𝑦 ∃𝑥 ∈ 𝐽 (𝑧 ∈ 𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝑦)))
2		sseq2 3992	. . . . . . . 8 ⊢ (𝑦 = 𝑈 → (((cls‘𝐽)‘𝑥) ⊆ 𝑦 ↔ ((cls‘𝐽)‘𝑥) ⊆ 𝑈))
3	2	anbi2d 630	. . . . . . 7 ⊢ (𝑦 = 𝑈 → ((𝑧 ∈ 𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝑦) ↔ (𝑧 ∈ 𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝑈)))
4	3	rexbidv 3297	. . . . . 6 ⊢ (𝑦 = 𝑈 → (∃𝑥 ∈ 𝐽 (𝑧 ∈ 𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝑦) ↔ ∃𝑥 ∈ 𝐽 (𝑧 ∈ 𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝑈)))
5	4	raleqbi1dv 3403	. . . . 5 ⊢ (𝑦 = 𝑈 → (∀𝑧 ∈ 𝑦 ∃𝑥 ∈ 𝐽 (𝑧 ∈ 𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝑦) ↔ ∀𝑧 ∈ 𝑈 ∃𝑥 ∈ 𝐽 (𝑧 ∈ 𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝑈)))
6	5	rspccv 3619	. . . 4 ⊢ (∀𝑦 ∈ 𝐽 ∀𝑧 ∈ 𝑦 ∃𝑥 ∈ 𝐽 (𝑧 ∈ 𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝑦) → (𝑈 ∈ 𝐽 → ∀𝑧 ∈ 𝑈 ∃𝑥 ∈ 𝐽 (𝑧 ∈ 𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝑈)))
7	1, 6	simplbiim 507	. . 3 ⊢ (𝐽 ∈ Reg → (𝑈 ∈ 𝐽 → ∀𝑧 ∈ 𝑈 ∃𝑥 ∈ 𝐽 (𝑧 ∈ 𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝑈)))
8		eleq1 2900	. . . . . 6 ⊢ (𝑧 = 𝐴 → (𝑧 ∈ 𝑥 ↔ 𝐴 ∈ 𝑥))
9	8	anbi1d 631	. . . . 5 ⊢ (𝑧 = 𝐴 → ((𝑧 ∈ 𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝑈) ↔ (𝐴 ∈ 𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝑈)))
10	9	rexbidv 3297	. . . 4 ⊢ (𝑧 = 𝐴 → (∃𝑥 ∈ 𝐽 (𝑧 ∈ 𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝑈) ↔ ∃𝑥 ∈ 𝐽 (𝐴 ∈ 𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝑈)))
11	10	rspccv 3619	. . 3 ⊢ (∀𝑧 ∈ 𝑈 ∃𝑥 ∈ 𝐽 (𝑧 ∈ 𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝑈) → (𝐴 ∈ 𝑈 → ∃𝑥 ∈ 𝐽 (𝐴 ∈ 𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝑈)))
12	7, 11	syl6 35	. 2 ⊢ (𝐽 ∈ Reg → (𝑈 ∈ 𝐽 → (𝐴 ∈ 𝑈 → ∃𝑥 ∈ 𝐽 (𝐴 ∈ 𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝑈))))
13	12	3imp 1107	1 ⊢ ((𝐽 ∈ Reg ∧ 𝑈 ∈ 𝐽 ∧ 𝐴 ∈ 𝑈) → ∃𝑥 ∈ 𝐽 (𝐴 ∈ 𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝑈))

Syntax hints:   → wi 4   ∧ wa 398   ∧ w3a 1083   = wceq 1533   ∈ wcel 2110  ∀wral 3138  ∃wrex 3139   ⊆ wss 3935  ‘cfv 6349  Topctop 21495  clsccl 21620  Regcreg 21911

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-br 5059  df-iota 6308  df-fv 6357  df-reg 21918

This theorem is referenced by:  regsep2  21978  regr1lem  22341  kqreglem1  22343  kqreglem2  22344  reghmph  22395  cnextcn  22669