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Mirrors > Home > MPE Home > Th. List > regsep | Structured version Visualization version GIF version |
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 23356 | . . . 4 ⊢ (𝐽 ∈ Reg ↔ (𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝐽 ∀𝑧 ∈ 𝑦 ∃𝑥 ∈ 𝐽 (𝑧 ∈ 𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝑦))) | |
2 | sseq2 4022 | . . . . . . . 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 3177 | . . . . . 6 ⊢ (𝑦 = 𝑈 → (∃𝑥 ∈ 𝐽 (𝑧 ∈ 𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝑦) ↔ ∃𝑥 ∈ 𝐽 (𝑧 ∈ 𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝑈))) |
5 | 4 | raleqbi1dv 3336 | . . . . 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 504 | . . 3 ⊢ (𝐽 ∈ Reg → (𝑈 ∈ 𝐽 → ∀𝑧 ∈ 𝑈 ∃𝑥 ∈ 𝐽 (𝑧 ∈ 𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝑈))) |
8 | eleq1 2827 | . . . . . 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 3177 | . . . 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 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 1537 ∈ wcel 2106 ∀wral 3059 ∃wrex 3068 ⊆ wss 3963 ‘cfv 6563 Topctop 22915 clsccl 23042 Regcreg 23333 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-iota 6516 df-fv 6571 df-reg 23340 |
This theorem is referenced by: regsep2 23400 regr1lem 23763 kqreglem1 23765 kqreglem2 23766 reghmph 23817 cnextcn 24091 |
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