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Mirrors > Home > MPE Home > Th. List > hauspwpwdom | Structured version Visualization version GIF version |
Description: If 𝑋 is a Hausdorff space, then the cardinality of the closure of a set 𝐴 is bounded by the double powerset of 𝐴. In particular, a Hausdorff space with a dense subset 𝐴 has cardinality at most 𝒫 𝒫 𝐴, and a separable Hausdorff space has cardinality at most 𝒫 𝒫 ℕ. (Contributed by Mario Carneiro, 9-Apr-2015.) (Revised by Mario Carneiro, 28-Jul-2015.) |
Ref | Expression |
---|---|
hauspwpwf1.x | ⊢ 𝑋 = ∪ 𝐽 |
Ref | Expression |
---|---|
hauspwpwdom | ⊢ ((𝐽 ∈ Haus ∧ 𝐴 ⊆ 𝑋) → ((cls‘𝐽)‘𝐴) ≼ 𝒫 𝒫 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvexd 6678 | . 2 ⊢ ((𝐽 ∈ Haus ∧ 𝐴 ⊆ 𝑋) → ((cls‘𝐽)‘𝐴) ∈ V) | |
2 | haustop 21867 | . . . . . 6 ⊢ (𝐽 ∈ Haus → 𝐽 ∈ Top) | |
3 | hauspwpwf1.x | . . . . . . 7 ⊢ 𝑋 = ∪ 𝐽 | |
4 | 3 | topopn 21442 | . . . . . 6 ⊢ (𝐽 ∈ Top → 𝑋 ∈ 𝐽) |
5 | 2, 4 | syl 17 | . . . . 5 ⊢ (𝐽 ∈ Haus → 𝑋 ∈ 𝐽) |
6 | 5 | adantr 481 | . . . 4 ⊢ ((𝐽 ∈ Haus ∧ 𝐴 ⊆ 𝑋) → 𝑋 ∈ 𝐽) |
7 | simpr 485 | . . . 4 ⊢ ((𝐽 ∈ Haus ∧ 𝐴 ⊆ 𝑋) → 𝐴 ⊆ 𝑋) | |
8 | 6, 7 | ssexd 5219 | . . 3 ⊢ ((𝐽 ∈ Haus ∧ 𝐴 ⊆ 𝑋) → 𝐴 ∈ V) |
9 | pwexg 5270 | . . 3 ⊢ (𝐴 ∈ V → 𝒫 𝐴 ∈ V) | |
10 | pwexg 5270 | . . 3 ⊢ (𝒫 𝐴 ∈ V → 𝒫 𝒫 𝐴 ∈ V) | |
11 | 8, 9, 10 | 3syl 18 | . 2 ⊢ ((𝐽 ∈ Haus ∧ 𝐴 ⊆ 𝑋) → 𝒫 𝒫 𝐴 ∈ V) |
12 | eqid 2818 | . . 3 ⊢ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ↦ {𝑧 ∣ ∃𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 ∧ 𝑧 = (𝑦 ∩ 𝐴))}) = (𝑥 ∈ ((cls‘𝐽)‘𝐴) ↦ {𝑧 ∣ ∃𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 ∧ 𝑧 = (𝑦 ∩ 𝐴))}) | |
13 | 3, 12 | hauspwpwf1 22523 | . 2 ⊢ ((𝐽 ∈ Haus ∧ 𝐴 ⊆ 𝑋) → (𝑥 ∈ ((cls‘𝐽)‘𝐴) ↦ {𝑧 ∣ ∃𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 ∧ 𝑧 = (𝑦 ∩ 𝐴))}):((cls‘𝐽)‘𝐴)–1-1→𝒫 𝒫 𝐴) |
14 | f1dom2g 8515 | . 2 ⊢ ((((cls‘𝐽)‘𝐴) ∈ V ∧ 𝒫 𝒫 𝐴 ∈ V ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ↦ {𝑧 ∣ ∃𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 ∧ 𝑧 = (𝑦 ∩ 𝐴))}):((cls‘𝐽)‘𝐴)–1-1→𝒫 𝒫 𝐴) → ((cls‘𝐽)‘𝐴) ≼ 𝒫 𝒫 𝐴) | |
15 | 1, 11, 13, 14 | syl3anc 1363 | 1 ⊢ ((𝐽 ∈ Haus ∧ 𝐴 ⊆ 𝑋) → ((cls‘𝐽)‘𝐴) ≼ 𝒫 𝒫 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 {cab 2796 ∃wrex 3136 Vcvv 3492 ∩ cin 3932 ⊆ wss 3933 𝒫 cpw 4535 ∪ cuni 4830 class class class wbr 5057 ↦ cmpt 5137 –1-1→wf1 6345 ‘cfv 6348 ≼ cdom 8495 Topctop 21429 clsccl 21554 Hauscha 21844 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-iin 4913 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-dom 8499 df-top 21430 df-cld 21555 df-ntr 21556 df-cls 21557 df-haus 21851 |
This theorem is referenced by: (None) |
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