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Theorem hauspwpwdom 23910
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 6907 . 2 ((𝐽 ∈ Haus ∧ 𝐴𝑋) → ((cls‘𝐽)‘𝐴) ∈ V)
2 haustop 23253 . . . . . 6 (𝐽 ∈ Haus → 𝐽 ∈ Top)
3 hauspwpwf1.x . . . . . . 7 𝑋 = 𝐽
43topopn 22826 . . . . . 6 (𝐽 ∈ Top → 𝑋𝐽)
52, 4syl 17 . . . . 5 (𝐽 ∈ Haus → 𝑋𝐽)
65adantr 479 . . . 4 ((𝐽 ∈ Haus ∧ 𝐴𝑋) → 𝑋𝐽)
7 simpr 483 . . . 4 ((𝐽 ∈ Haus ∧ 𝐴𝑋) → 𝐴𝑋)
86, 7ssexd 5319 . . 3 ((𝐽 ∈ Haus ∧ 𝐴𝑋) → 𝐴 ∈ V)
9 pwexg 5372 . . 3 (𝐴 ∈ V → 𝒫 𝐴 ∈ V)
10 pwexg 5372 . . 3 (𝒫 𝐴 ∈ V → 𝒫 𝒫 𝐴 ∈ V)
118, 9, 103syl 18 . 2 ((𝐽 ∈ Haus ∧ 𝐴𝑋) → 𝒫 𝒫 𝐴 ∈ V)
12 eqid 2725 . . 3 (𝑥 ∈ ((cls‘𝐽)‘𝐴) ↦ {𝑧 ∣ ∃𝑦𝐽 (𝑥𝑦𝑧 = (𝑦𝐴))}) = (𝑥 ∈ ((cls‘𝐽)‘𝐴) ↦ {𝑧 ∣ ∃𝑦𝐽 (𝑥𝑦𝑧 = (𝑦𝐴))})
133, 12hauspwpwf1 23909 . 2 ((𝐽 ∈ Haus ∧ 𝐴𝑋) → (𝑥 ∈ ((cls‘𝐽)‘𝐴) ↦ {𝑧 ∣ ∃𝑦𝐽 (𝑥𝑦𝑧 = (𝑦𝐴))}):((cls‘𝐽)‘𝐴)–1-1→𝒫 𝒫 𝐴)
14 f1dom2g 8988 . 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 1368 1 ((𝐽 ∈ Haus ∧ 𝐴𝑋) → ((cls‘𝐽)‘𝐴) ≼ 𝒫 𝒫 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1533  wcel 2098  {cab 2702  wrex 3060  Vcvv 3463  cin 3938  wss 3939  𝒫 cpw 4598   cuni 4903   class class class wbr 5143  cmpt 5226  1-1wf1 6540  cfv 6543  cdom 8960  Topctop 22813  clsccl 22940  Hauscha 23230
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7738
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-dom 8964  df-top 22814  df-cld 22941  df-ntr 22942  df-cls 22943  df-haus 23237
This theorem is referenced by: (None)
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