Theorem regsep 23363

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 23361	. . . 4 ⊢ (𝐽 ∈ Reg ↔ (𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝐽 ∀𝑧 ∈ 𝑦 ∃𝑥 ∈ 𝐽 (𝑧 ∈ 𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝑦)))
2		sseq2 4035	. . . . . . . 8 ⊢ (𝑦 = 𝑈 → (((cls‘𝐽)‘𝑥) ⊆ 𝑦 ↔ ((cls‘𝐽)‘𝑥) ⊆ 𝑈))
3	2	anbi2d 629	. . . . . . 7 ⊢ (𝑦 = 𝑈 → ((𝑧 ∈ 𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝑦) ↔ (𝑧 ∈ 𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝑈)))
4	3	rexbidv 3185	. . . . . 6 ⊢ (𝑦 = 𝑈 → (∃𝑥 ∈ 𝐽 (𝑧 ∈ 𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝑦) ↔ ∃𝑥 ∈ 𝐽 (𝑧 ∈ 𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝑈)))
5	4	raleqbi1dv 3346	. . . . 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 3632	. . . 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 504	. . 3 ⊢ (𝐽 ∈ Reg → (𝑈 ∈ 𝐽 → ∀𝑧 ∈ 𝑈 ∃𝑥 ∈ 𝐽 (𝑧 ∈ 𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝑈)))
8		eleq1 2832	. . . . . 6 ⊢ (𝑧 = 𝐴 → (𝑧 ∈ 𝑥 ↔ 𝐴 ∈ 𝑥))
9	8	anbi1d 630	. . . . 5 ⊢ (𝑧 = 𝐴 → ((𝑧 ∈ 𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝑈) ↔ (𝐴 ∈ 𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝑈)))
10	9	rexbidv 3185	. . . 4 ⊢ (𝑧 = 𝐴 → (∃𝑥 ∈ 𝐽 (𝑧 ∈ 𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝑈) ↔ ∃𝑥 ∈ 𝐽 (𝐴 ∈ 𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝑈)))
11	10	rspccv 3632	. . 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 1111	1 ⊢ ((𝐽 ∈ Reg ∧ 𝑈 ∈ 𝐽 ∧ 𝐴 ∈ 𝑈) → ∃𝑥 ∈ 𝐽 (𝐴 ∈ 𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝑈))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108  ∀wral 3067  ∃wrex 3076   ⊆ wss 3976  ‘cfv 6573  Topctop 22920  clsccl 23047  Regcreg 23338

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-iota 6525  df-fv 6581  df-reg 23345

This theorem is referenced by:  regsep2  23405  regr1lem  23768  kqreglem1  23770  kqreglem2  23771  reghmph  23822  cnextcn  24096