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| Description: In a regular space, every neighborhood of a point contains a closed subneighborhood. (Contributed by Mario Carneiro, 25-Aug-2015.) | 
| Ref | Expression | 
|---|---|
| regsep | ⊢ ((𝐽 ∈ Reg ∧ 𝑈 ∈ 𝐽 ∧ 𝐴 ∈ 𝑈) → ∃𝑥 ∈ 𝐽 (𝐴 ∈ 𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝑈)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | isreg 23341 | . . . 4 ⊢ (𝐽 ∈ Reg ↔ (𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝐽 ∀𝑧 ∈ 𝑦 ∃𝑥 ∈ 𝐽 (𝑧 ∈ 𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝑦))) | |
| 2 | sseq2 4009 | . . . . . . . 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 3178 | . . . . . 6 ⊢ (𝑦 = 𝑈 → (∃𝑥 ∈ 𝐽 (𝑧 ∈ 𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝑦) ↔ ∃𝑥 ∈ 𝐽 (𝑧 ∈ 𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝑈))) | 
| 5 | 4 | raleqbi1dv 3337 | . . . . 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 3618 | . . . 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 2828 | . . . . . 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 3178 | . . . 4 ⊢ (𝑧 = 𝐴 → (∃𝑥 ∈ 𝐽 (𝑧 ∈ 𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝑈) ↔ ∃𝑥 ∈ 𝐽 (𝐴 ∈ 𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝑈))) | 
| 11 | 10 | rspccv 3618 | . . 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 1110 | 1 ⊢ ((𝐽 ∈ Reg ∧ 𝑈 ∈ 𝐽 ∧ 𝐴 ∈ 𝑈) → ∃𝑥 ∈ 𝐽 (𝐴 ∈ 𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝑈)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1539 ∈ wcel 2107 ∀wral 3060 ∃wrex 3069 ⊆ wss 3950 ‘cfv 6560 Topctop 22900 clsccl 23027 Regcreg 23318 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-iota 6513 df-fv 6568 df-reg 23325 | 
| This theorem is referenced by: regsep2 23385 regr1lem 23748 kqreglem1 23750 kqreglem2 23751 reghmph 23802 cnextcn 24076 | 
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