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Mirrors > Home > MPE Home > Th. List > isreg | Structured version Visualization version GIF version |
Description: The predicate "is a regular space". In a regular space, any open neighborhood has a closed subneighborhood. Note that some authors require the space to be Hausdorff (which would make it the same as T3), but we reserve the phrase "regular Hausdorff" for that as many topologists do. (Contributed by Jeff Hankins, 1-Feb-2010.) (Revised by Mario Carneiro, 25-Aug-2015.) |
Ref | Expression |
---|---|
isreg | ⊢ (𝐽 ∈ Reg ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ 𝐽 (𝑦 ∈ 𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑥))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6892 | . . . . . . . 8 ⊢ (𝑗 = 𝐽 → (cls‘𝑗) = (cls‘𝐽)) | |
2 | 1 | fveq1d 6894 | . . . . . . 7 ⊢ (𝑗 = 𝐽 → ((cls‘𝑗)‘𝑧) = ((cls‘𝐽)‘𝑧)) |
3 | 2 | sseq1d 4014 | . . . . . 6 ⊢ (𝑗 = 𝐽 → (((cls‘𝑗)‘𝑧) ⊆ 𝑥 ↔ ((cls‘𝐽)‘𝑧) ⊆ 𝑥)) |
4 | 3 | anbi2d 630 | . . . . 5 ⊢ (𝑗 = 𝐽 → ((𝑦 ∈ 𝑧 ∧ ((cls‘𝑗)‘𝑧) ⊆ 𝑥) ↔ (𝑦 ∈ 𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑥))) |
5 | 4 | rexeqbi1dv 3335 | . . . 4 ⊢ (𝑗 = 𝐽 → (∃𝑧 ∈ 𝑗 (𝑦 ∈ 𝑧 ∧ ((cls‘𝑗)‘𝑧) ⊆ 𝑥) ↔ ∃𝑧 ∈ 𝐽 (𝑦 ∈ 𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑥))) |
6 | 5 | ralbidv 3178 | . . 3 ⊢ (𝑗 = 𝐽 → (∀𝑦 ∈ 𝑥 ∃𝑧 ∈ 𝑗 (𝑦 ∈ 𝑧 ∧ ((cls‘𝑗)‘𝑧) ⊆ 𝑥) ↔ ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ 𝐽 (𝑦 ∈ 𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑥))) |
7 | 6 | raleqbi1dv 3334 | . 2 ⊢ (𝑗 = 𝐽 → (∀𝑥 ∈ 𝑗 ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ 𝑗 (𝑦 ∈ 𝑧 ∧ ((cls‘𝑗)‘𝑧) ⊆ 𝑥) ↔ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ 𝐽 (𝑦 ∈ 𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑥))) |
8 | df-reg 22820 | . 2 ⊢ Reg = {𝑗 ∈ Top ∣ ∀𝑥 ∈ 𝑗 ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ 𝑗 (𝑦 ∈ 𝑧 ∧ ((cls‘𝑗)‘𝑧) ⊆ 𝑥)} | |
9 | 7, 8 | elrab2 3687 | 1 ⊢ (𝐽 ∈ Reg ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ 𝐽 (𝑦 ∈ 𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑥))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∀wral 3062 ∃wrex 3071 ⊆ wss 3949 ‘cfv 6544 Topctop 22395 clsccl 22522 Regcreg 22813 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-iota 6496 df-fv 6552 df-reg 22820 |
This theorem is referenced by: regtop 22837 regsep 22838 isreg2 22881 kqreglem1 23245 kqreglem2 23246 nrmr0reg 23253 reghmph 23297 utopreg 23757 |
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