Theorem hauspwpwdom 23847

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 6900	. 2 ⊢ ((𝐽 ∈ Haus ∧ 𝐴 ⊆ 𝑋) → ((cls‘𝐽)‘𝐴) ∈ V)
2		haustop 23190	. . . . . 6 ⊢ (𝐽 ∈ Haus → 𝐽 ∈ Top)
3		hauspwpwf1.x	. . . . . . 7 ⊢ 𝑋 = ∪ 𝐽
4	3	topopn 22763	. . . . . 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 5317	. . 3 ⊢ ((𝐽 ∈ Haus ∧ 𝐴 ⊆ 𝑋) → 𝐴 ∈ V)
9		pwexg 5369	. . 3 ⊢ (𝐴 ∈ V → 𝒫 𝐴 ∈ V)
10		pwexg 5369	. . 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 2726	. . 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 23846	. 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 8967	. 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 1368	1 ⊢ ((𝐽 ∈ Haus ∧ 𝐴 ⊆ 𝑋) → ((cls‘𝐽)‘𝐴) ≼ 𝒫 𝒫 𝐴)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098  {cab 2703  ∃wrex 3064  Vcvv 3468   ∩ cin 3942   ⊆ wss 3943  𝒫 cpw 4597  ∪ cuni 4902   class class class wbr 5141   ↦ cmpt 5224  –1-1→wf1 6534  ‘cfv 6537   ≼ cdom 8939  Topctop 22750  clsccl 22877  Hauscha 23167

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 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  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 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-int 4944  df-iun 4992  df-iin 4993  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-dom 8943  df-top 22751  df-cld 22878  df-ntr 22879  df-cls 22880  df-haus 23174

This theorem is referenced by: (None)