Theorem hauspwpwdom 23944

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 6857	. 2 ⊢ ((𝐽 ∈ Haus ∧ 𝐴 ⊆ 𝑋) → ((cls‘𝐽)‘𝐴) ∈ V)
2		haustop 23287	. . . . . 6 ⊢ (𝐽 ∈ Haus → 𝐽 ∈ Top)
3		hauspwpwf1.x	. . . . . . 7 ⊢ 𝑋 = ∪ 𝐽
4	3	topopn 22862	. . . . . 6 ⊢ (𝐽 ∈ Top → 𝑋 ∈ 𝐽)
5	2, 4	syl 17	. . . . 5 ⊢ (𝐽 ∈ Haus → 𝑋 ∈ 𝐽)
6	5	adantr 480	. . . 4 ⊢ ((𝐽 ∈ Haus ∧ 𝐴 ⊆ 𝑋) → 𝑋 ∈ 𝐽)
7		simpr 484	. . . 4 ⊢ ((𝐽 ∈ Haus ∧ 𝐴 ⊆ 𝑋) → 𝐴 ⊆ 𝑋)
8	6, 7	ssexd 5271	. . 3 ⊢ ((𝐽 ∈ Haus ∧ 𝐴 ⊆ 𝑋) → 𝐴 ∈ V)
9		pwexg 5325	. . 3 ⊢ (𝐴 ∈ V → 𝒫 𝐴 ∈ V)
10		pwexg 5325	. . 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 2737	. . 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 23943	. 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 8918	. 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 1374	1 ⊢ ((𝐽 ∈ Haus ∧ 𝐴 ⊆ 𝑋) → ((cls‘𝐽)‘𝐴) ≼ 𝒫 𝒫 𝐴)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  {cab 2715  ∃wrex 3062  Vcvv 3442   ∩ cin 3902   ⊆ wss 3903  𝒫 cpw 4556  ∪ cuni 4865   class class class wbr 5100   ↦ cmpt 5181  –1-1→wf1 6497  ‘cfv 6500   ≼ cdom 8893  Topctop 22849  clsccl 22974  Hauscha 23264

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4905  df-iun 4950  df-iin 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-dom 8897  df-top 22850  df-cld 22975  df-ntr 22976  df-cls 22977  df-haus 23271

This theorem is referenced by: (None)