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Theorem hauspwpwdom 23996
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 6921 . 2 ((𝐽 ∈ Haus ∧ 𝐴𝑋) → ((cls‘𝐽)‘𝐴) ∈ V)
2 haustop 23339 . . . . . 6 (𝐽 ∈ Haus → 𝐽 ∈ Top)
3 hauspwpwf1.x . . . . . . 7 𝑋 = 𝐽
43topopn 22912 . . . . . 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 5324 . . 3 ((𝐽 ∈ Haus ∧ 𝐴𝑋) → 𝐴 ∈ V)
9 pwexg 5378 . . 3 (𝐴 ∈ V → 𝒫 𝐴 ∈ V)
10 pwexg 5378 . . 3 (𝒫 𝐴 ∈ V → 𝒫 𝒫 𝐴 ∈ V)
118, 9, 103syl 18 . 2 ((𝐽 ∈ Haus ∧ 𝐴𝑋) → 𝒫 𝒫 𝐴 ∈ V)
12 eqid 2737 . . 3 (𝑥 ∈ ((cls‘𝐽)‘𝐴) ↦ {𝑧 ∣ ∃𝑦𝐽 (𝑥𝑦𝑧 = (𝑦𝐴))}) = (𝑥 ∈ ((cls‘𝐽)‘𝐴) ↦ {𝑧 ∣ ∃𝑦𝐽 (𝑥𝑦𝑧 = (𝑦𝐴))})
133, 12hauspwpwf1 23995 . 2 ((𝐽 ∈ Haus ∧ 𝐴𝑋) → (𝑥 ∈ ((cls‘𝐽)‘𝐴) ↦ {𝑧 ∣ ∃𝑦𝐽 (𝑥𝑦𝑧 = (𝑦𝐴))}):((cls‘𝐽)‘𝐴)–1-1→𝒫 𝒫 𝐴)
14 f1dom2g 9010 . 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 1373 1 ((𝐽 ∈ Haus ∧ 𝐴𝑋) → ((cls‘𝐽)‘𝐴) ≼ 𝒫 𝒫 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  {cab 2714  wrex 3070  Vcvv 3480  cin 3950  wss 3951  𝒫 cpw 4600   cuni 4907   class class class wbr 5143  cmpt 5225  1-1wf1 6558  cfv 6561  cdom 8983  Topctop 22899  clsccl 23026  Hauscha 23316
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-dom 8987  df-top 22900  df-cld 23027  df-ntr 23028  df-cls 23029  df-haus 23323
This theorem is referenced by: (None)
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