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Theorem hauspwpwdom 23362
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 6861 . 2 ((𝐽 ∈ Haus ∧ 𝐴𝑋) → ((cls‘𝐽)‘𝐴) ∈ V)
2 haustop 22705 . . . . . 6 (𝐽 ∈ Haus → 𝐽 ∈ Top)
3 hauspwpwf1.x . . . . . . 7 𝑋 = 𝐽
43topopn 22278 . . . . . 6 (𝐽 ∈ Top → 𝑋𝐽)
52, 4syl 17 . . . . 5 (𝐽 ∈ Haus → 𝑋𝐽)
65adantr 482 . . . 4 ((𝐽 ∈ Haus ∧ 𝐴𝑋) → 𝑋𝐽)
7 simpr 486 . . . 4 ((𝐽 ∈ Haus ∧ 𝐴𝑋) → 𝐴𝑋)
86, 7ssexd 5285 . . 3 ((𝐽 ∈ Haus ∧ 𝐴𝑋) → 𝐴 ∈ V)
9 pwexg 5337 . . 3 (𝐴 ∈ V → 𝒫 𝐴 ∈ V)
10 pwexg 5337 . . 3 (𝒫 𝐴 ∈ V → 𝒫 𝒫 𝐴 ∈ V)
118, 9, 103syl 18 . 2 ((𝐽 ∈ Haus ∧ 𝐴𝑋) → 𝒫 𝒫 𝐴 ∈ V)
12 eqid 2733 . . 3 (𝑥 ∈ ((cls‘𝐽)‘𝐴) ↦ {𝑧 ∣ ∃𝑦𝐽 (𝑥𝑦𝑧 = (𝑦𝐴))}) = (𝑥 ∈ ((cls‘𝐽)‘𝐴) ↦ {𝑧 ∣ ∃𝑦𝐽 (𝑥𝑦𝑧 = (𝑦𝐴))})
133, 12hauspwpwf1 23361 . 2 ((𝐽 ∈ Haus ∧ 𝐴𝑋) → (𝑥 ∈ ((cls‘𝐽)‘𝐴) ↦ {𝑧 ∣ ∃𝑦𝐽 (𝑥𝑦𝑧 = (𝑦𝐴))}):((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 ∣ ∃𝑦𝐽 (𝑥𝑦𝑧 = (𝑦𝐴))}):((cls‘𝐽)‘𝐴)–1-1→𝒫 𝒫 𝐴) → ((cls‘𝐽)‘𝐴) ≼ 𝒫 𝒫 𝐴)
151, 11, 13, 14syl3anc 1372 1 ((𝐽 ∈ Haus ∧ 𝐴𝑋) → ((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  1-1wf1 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)
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