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Mirrors > Home > MPE Home > Th. List > pnrmcld | Structured version Visualization version GIF version |
Description: A closed set in a perfectly normal space is a countable intersection of open sets. (Contributed by Mario Carneiro, 26-Aug-2015.) |
Ref | Expression |
---|---|
pnrmcld | ⊢ ((𝐽 ∈ PNrm ∧ 𝐴 ∈ (Clsd‘𝐽)) → ∃𝑓 ∈ (𝐽 ↑m ℕ)𝐴 = ∩ ran 𝑓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ispnrm 22083 | . . . 4 ⊢ (𝐽 ∈ PNrm ↔ (𝐽 ∈ Nrm ∧ (Clsd‘𝐽) ⊆ ran (𝑓 ∈ (𝐽 ↑m ℕ) ↦ ∩ ran 𝑓))) | |
2 | 1 | simprbi 500 | . . 3 ⊢ (𝐽 ∈ PNrm → (Clsd‘𝐽) ⊆ ran (𝑓 ∈ (𝐽 ↑m ℕ) ↦ ∩ ran 𝑓)) |
3 | 2 | sselda 3875 | . 2 ⊢ ((𝐽 ∈ PNrm ∧ 𝐴 ∈ (Clsd‘𝐽)) → 𝐴 ∈ ran (𝑓 ∈ (𝐽 ↑m ℕ) ↦ ∩ ran 𝑓)) |
4 | eqid 2738 | . . . 4 ⊢ (𝑓 ∈ (𝐽 ↑m ℕ) ↦ ∩ ran 𝑓) = (𝑓 ∈ (𝐽 ↑m ℕ) ↦ ∩ ran 𝑓) | |
5 | 4 | elrnmpt 5793 | . . 3 ⊢ (𝐴 ∈ (Clsd‘𝐽) → (𝐴 ∈ ran (𝑓 ∈ (𝐽 ↑m ℕ) ↦ ∩ ran 𝑓) ↔ ∃𝑓 ∈ (𝐽 ↑m ℕ)𝐴 = ∩ ran 𝑓)) |
6 | 5 | adantl 485 | . 2 ⊢ ((𝐽 ∈ PNrm ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝐴 ∈ ran (𝑓 ∈ (𝐽 ↑m ℕ) ↦ ∩ ran 𝑓) ↔ ∃𝑓 ∈ (𝐽 ↑m ℕ)𝐴 = ∩ ran 𝑓)) |
7 | 3, 6 | mpbid 235 | 1 ⊢ ((𝐽 ∈ PNrm ∧ 𝐴 ∈ (Clsd‘𝐽)) → ∃𝑓 ∈ (𝐽 ↑m ℕ)𝐴 = ∩ ran 𝑓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1542 ∈ wcel 2113 ∃wrex 3054 ⊆ wss 3841 ∩ cint 4833 ↦ cmpt 5107 ran crn 5520 ‘cfv 6333 (class class class)co 7164 ↑m cmap 8430 ℕcn 11709 Clsdccld 21760 Nrmcnrm 22054 PNrmcpnrm 22056 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-sep 5164 ax-nul 5171 ax-pr 5293 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ral 3058 df-rex 3059 df-rab 3062 df-v 3399 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-nul 4210 df-if 4412 df-sn 4514 df-pr 4516 df-op 4520 df-uni 4794 df-br 5028 df-opab 5090 df-mpt 5108 df-cnv 5527 df-dm 5529 df-rn 5530 df-iota 6291 df-fv 6341 df-ov 7167 df-pnrm 22063 |
This theorem is referenced by: pnrmopn 22087 |
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