![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 6861 | . 2 ⊢ ((𝐽 ∈ Haus ∧ 𝐴 ⊆ 𝑋) → ((cls‘𝐽)‘𝐴) ∈ V) | |
2 | haustop 22705 | . . . . . 6 ⊢ (𝐽 ∈ Haus → 𝐽 ∈ Top) | |
3 | hauspwpwf1.x | . . . . . . 7 ⊢ 𝑋 = ∪ 𝐽 | |
4 | 3 | topopn 22278 | . . . . . 6 ⊢ (𝐽 ∈ Top → 𝑋 ∈ 𝐽) |
5 | 2, 4 | syl 17 | . . . . 5 ⊢ (𝐽 ∈ Haus → 𝑋 ∈ 𝐽) |
6 | 5 | adantr 482 | . . . 4 ⊢ ((𝐽 ∈ Haus ∧ 𝐴 ⊆ 𝑋) → 𝑋 ∈ 𝐽) |
7 | simpr 486 | . . . 4 ⊢ ((𝐽 ∈ Haus ∧ 𝐴 ⊆ 𝑋) → 𝐴 ⊆ 𝑋) | |
8 | 6, 7 | ssexd 5285 | . . 3 ⊢ ((𝐽 ∈ Haus ∧ 𝐴 ⊆ 𝑋) → 𝐴 ∈ V) |
9 | pwexg 5337 | . . 3 ⊢ (𝐴 ∈ V → 𝒫 𝐴 ∈ V) | |
10 | pwexg 5337 | . . 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 2733 | . . 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 23361 | . 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 8915 | . 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 1372 | 1 ⊢ ((𝐽 ∈ Haus ∧ 𝐴 ⊆ 𝑋) → ((cls‘𝐽)‘𝐴) ≼ 𝒫 𝒫 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 {cab 2710 ∃wrex 3070 Vcvv 3447 ∩ cin 3913 ⊆ wss 3914 𝒫 cpw 4564 ∪ cuni 4869 class class class wbr 5109 ↦ cmpt 5192 –1-1→wf1 6497 ‘cfv 6500 ≼ cdom 8887 Topctop 22265 clsccl 22392 Hauscha 22682 |
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-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5246 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 |
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-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-int 4912 df-iun 4960 df-iin 4961 df-br 5110 df-opab 5172 df-mpt 5193 df-id 5535 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-dom 8891 df-top 22266 df-cld 22393 df-ntr 22394 df-cls 22395 df-haus 22689 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |