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Theorem hauspwpwdom 24012
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.)
Hypothesis
Ref Expression
hauspwpwf1.x 𝑋 = 𝐽
Assertion
Ref Expression
hauspwpwdom ((𝐽 ∈ Haus ∧ 𝐴𝑋) → ((cls‘𝐽)‘𝐴) ≼ 𝒫 𝒫 𝐴)

Proof of Theorem hauspwpwdom
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fvexd 6922 . 2 ((𝐽 ∈ Haus ∧ 𝐴𝑋) → ((cls‘𝐽)‘𝐴) ∈ V)
2 haustop 23355 . . . . . 6 (𝐽 ∈ Haus → 𝐽 ∈ Top)
3 hauspwpwf1.x . . . . . . 7 𝑋 = 𝐽
43topopn 22928 . . . . . 6 (𝐽 ∈ Top → 𝑋𝐽)
52, 4syl 17 . . . . 5 (𝐽 ∈ Haus → 𝑋𝐽)
65adantr 480 . . . 4 ((𝐽 ∈ Haus ∧ 𝐴𝑋) → 𝑋𝐽)
7 simpr 484 . . . 4 ((𝐽 ∈ Haus ∧ 𝐴𝑋) → 𝐴𝑋)
86, 7ssexd 5330 . . 3 ((𝐽 ∈ Haus ∧ 𝐴𝑋) → 𝐴 ∈ V)
9 pwexg 5384 . . 3 (𝐴 ∈ V → 𝒫 𝐴 ∈ V)
10 pwexg 5384 . . 3 (𝒫 𝐴 ∈ V → 𝒫 𝒫 𝐴 ∈ V)
118, 9, 103syl 18 . 2 ((𝐽 ∈ Haus ∧ 𝐴𝑋) → 𝒫 𝒫 𝐴 ∈ V)
12 eqid 2735 . . 3 (𝑥 ∈ ((cls‘𝐽)‘𝐴) ↦ {𝑧 ∣ ∃𝑦𝐽 (𝑥𝑦𝑧 = (𝑦𝐴))}) = (𝑥 ∈ ((cls‘𝐽)‘𝐴) ↦ {𝑧 ∣ ∃𝑦𝐽 (𝑥𝑦𝑧 = (𝑦𝐴))})
133, 12hauspwpwf1 24011 . 2 ((𝐽 ∈ Haus ∧ 𝐴𝑋) → (𝑥 ∈ ((cls‘𝐽)‘𝐴) ↦ {𝑧 ∣ ∃𝑦𝐽 (𝑥𝑦𝑧 = (𝑦𝐴))}):((cls‘𝐽)‘𝐴)–1-1→𝒫 𝒫 𝐴)
14 f1dom2g 9009 . 2 ((((cls‘𝐽)‘𝐴) ∈ V ∧ 𝒫 𝒫 𝐴 ∈ V ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ↦ {𝑧 ∣ ∃𝑦𝐽 (𝑥𝑦𝑧 = (𝑦𝐴))}):((cls‘𝐽)‘𝐴)–1-1→𝒫 𝒫 𝐴) → ((cls‘𝐽)‘𝐴) ≼ 𝒫 𝒫 𝐴)
151, 11, 13, 14syl3anc 1370 1 ((𝐽 ∈ Haus ∧ 𝐴𝑋) → ((cls‘𝐽)‘𝐴) ≼ 𝒫 𝒫 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  {cab 2712  wrex 3068  Vcvv 3478  cin 3962  wss 3963  𝒫 cpw 4605   cuni 4912   class class class wbr 5148  cmpt 5231  1-1wf1 6560  cfv 6563  cdom 8982  Topctop 22915  clsccl 23042  Hauscha 23332
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-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
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-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-dom 8986  df-top 22916  df-cld 23043  df-ntr 23044  df-cls 23045  df-haus 23339
This theorem is referenced by: (None)
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