Theorem hauspwpwdom 23139

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 ∧ 𝐴 ⊆ 𝑋) → ((cls‘𝐽)‘𝐴) ≼ 𝒫 𝒫 𝐴)

Proof of Theorem hauspwpwdom

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fvexd 6789	. 2 ⊢ ((𝐽 ∈ Haus ∧ 𝐴 ⊆ 𝑋) → ((cls‘𝐽)‘𝐴) ∈ V)
2		haustop 22482	. . . . . 6 ⊢ (𝐽 ∈ Haus → 𝐽 ∈ Top)
3		hauspwpwf1.x	. . . . . . 7 ⊢ 𝑋 = ∪ 𝐽
4	3	topopn 22055	. . . . . 6 ⊢ (𝐽 ∈ Top → 𝑋 ∈ 𝐽)
5	2, 4	syl 17	. . . . 5 ⊢ (𝐽 ∈ Haus → 𝑋 ∈ 𝐽)
6	5	adantr 481	. . . 4 ⊢ ((𝐽 ∈ Haus ∧ 𝐴 ⊆ 𝑋) → 𝑋 ∈ 𝐽)
7		simpr 485	. . . 4 ⊢ ((𝐽 ∈ Haus ∧ 𝐴 ⊆ 𝑋) → 𝐴 ⊆ 𝑋)
8	6, 7	ssexd 5248	. . 3 ⊢ ((𝐽 ∈ Haus ∧ 𝐴 ⊆ 𝑋) → 𝐴 ∈ V)
9		pwexg 5301	. . 3 ⊢ (𝐴 ∈ V → 𝒫 𝐴 ∈ V)
10		pwexg 5301	. . 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 2738	. . 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 23138	. 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 8757	. 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 1370	1 ⊢ ((𝐽 ∈ Haus ∧ 𝐴 ⊆ 𝑋) → ((cls‘𝐽)‘𝐴) ≼ 𝒫 𝒫 𝐴)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1539   ∈ wcel 2106  {cab 2715  ∃wrex 3065  Vcvv 3432   ∩ cin 3886   ⊆ wss 3887  𝒫 cpw 4533  ∪ cuni 4839   class class class wbr 5074   ↦ cmpt 5157  –1-1→wf1 6430  ‘cfv 6433   ≼ cdom 8731  Topctop 22042  clsccl 22169  Hauscha 22459

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-iin 4927  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-dom 8735  df-top 22043  df-cld 22170  df-ntr 22171  df-cls 22172  df-haus 22466

This theorem is referenced by: (None)