Theorem hausmapdom 22651

Description: If 𝑋 is a first-countable Hausdorff space, then the cardinality of the closure of a set 𝐴 is bounded by ℕ to the power 𝐴. In particular, a first-countable Hausdorff space with a dense subset 𝐴 has cardinality at most 𝐴↑ℕ, and a separable first-countable Hausdorff space has cardinality at most 𝒫 ℕ. (Compare hauspwpwdom 23139 to see a weaker result if the assumption of first-countability is omitted.) (Contributed by Mario Carneiro, 9-Apr-2015.)

Hypothesis

Ref	Expression
hauspwdom.1	⊢ 𝑋 = ∪ 𝐽

Assertion

Ref	Expression
hausmapdom	⊢ ((𝐽 ∈ Haus ∧ 𝐽 ∈ 1stω ∧ 𝐴 ⊆ 𝑋) → ((cls‘𝐽)‘𝐴) ≼ (𝐴 ↑m ℕ))

Proof of Theorem hausmapdom

Dummy variables 𝑥 𝑓 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		hauspwdom.1	. . . . . . . 8 ⊢ 𝑋 = ∪ 𝐽
2	1	1stcelcls 22612	. . . . . . 7 ⊢ ((𝐽 ∈ 1stω ∧ 𝐴 ⊆ 𝑋) → (𝑥 ∈ ((cls‘𝐽)‘𝐴) ↔ ∃𝑓(𝑓:ℕ⟶𝐴 ∧ 𝑓(⇝𝑡‘𝐽)𝑥)))
3	2	3adant1 1129	. . . . . 6 ⊢ ((𝐽 ∈ Haus ∧ 𝐽 ∈ 1stω ∧ 𝐴 ⊆ 𝑋) → (𝑥 ∈ ((cls‘𝐽)‘𝐴) ↔ ∃𝑓(𝑓:ℕ⟶𝐴 ∧ 𝑓(⇝𝑡‘𝐽)𝑥)))
4		uniexg 7593	. . . . . . . . . . . 12 ⊢ (𝐽 ∈ Haus → ∪ 𝐽 ∈ V)
5	4	3ad2ant1 1132	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ Haus ∧ 𝐽 ∈ 1stω ∧ 𝐴 ⊆ 𝑋) → ∪ 𝐽 ∈ V)
6	1, 5	eqeltrid 2843	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Haus ∧ 𝐽 ∈ 1stω ∧ 𝐴 ⊆ 𝑋) → 𝑋 ∈ V)
7		simp3 1137	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Haus ∧ 𝐽 ∈ 1stω ∧ 𝐴 ⊆ 𝑋) → 𝐴 ⊆ 𝑋)
8	6, 7	ssexd 5248	. . . . . . . . 9 ⊢ ((𝐽 ∈ Haus ∧ 𝐽 ∈ 1stω ∧ 𝐴 ⊆ 𝑋) → 𝐴 ∈ V)
9		nnex 11979	. . . . . . . . 9 ⊢ ℕ ∈ V
10		elmapg 8628	. . . . . . . . 9 ⊢ ((𝐴 ∈ V ∧ ℕ ∈ V) → (𝑓 ∈ (𝐴 ↑m ℕ) ↔ 𝑓:ℕ⟶𝐴))
11	8, 9, 10	sylancl 586	. . . . . . . 8 ⊢ ((𝐽 ∈ Haus ∧ 𝐽 ∈ 1stω ∧ 𝐴 ⊆ 𝑋) → (𝑓 ∈ (𝐴 ↑m ℕ) ↔ 𝑓:ℕ⟶𝐴))
12	11	anbi1d 630	. . . . . . 7 ⊢ ((𝐽 ∈ Haus ∧ 𝐽 ∈ 1stω ∧ 𝐴 ⊆ 𝑋) → ((𝑓 ∈ (𝐴 ↑m ℕ) ∧ 𝑓(⇝𝑡‘𝐽)𝑥) ↔ (𝑓:ℕ⟶𝐴 ∧ 𝑓(⇝𝑡‘𝐽)𝑥)))
13	12	exbidv 1924	. . . . . 6 ⊢ ((𝐽 ∈ Haus ∧ 𝐽 ∈ 1stω ∧ 𝐴 ⊆ 𝑋) → (∃𝑓(𝑓 ∈ (𝐴 ↑m ℕ) ∧ 𝑓(⇝𝑡‘𝐽)𝑥) ↔ ∃𝑓(𝑓:ℕ⟶𝐴 ∧ 𝑓(⇝𝑡‘𝐽)𝑥)))
14	3, 13	bitr4d 281	. . . . 5 ⊢ ((𝐽 ∈ Haus ∧ 𝐽 ∈ 1stω ∧ 𝐴 ⊆ 𝑋) → (𝑥 ∈ ((cls‘𝐽)‘𝐴) ↔ ∃𝑓(𝑓 ∈ (𝐴 ↑m ℕ) ∧ 𝑓(⇝𝑡‘𝐽)𝑥)))
15		df-rex 3070	. . . . 5 ⊢ (∃𝑓 ∈ (𝐴 ↑m ℕ)𝑓(⇝𝑡‘𝐽)𝑥 ↔ ∃𝑓(𝑓 ∈ (𝐴 ↑m ℕ) ∧ 𝑓(⇝𝑡‘𝐽)𝑥))
16	14, 15	bitr4di 289	. . . 4 ⊢ ((𝐽 ∈ Haus ∧ 𝐽 ∈ 1stω ∧ 𝐴 ⊆ 𝑋) → (𝑥 ∈ ((cls‘𝐽)‘𝐴) ↔ ∃𝑓 ∈ (𝐴 ↑m ℕ)𝑓(⇝𝑡‘𝐽)𝑥))
17		vex 3436	. . . . 5 ⊢ 𝑥 ∈ V
18	17	elima 5974	. . . 4 ⊢ (𝑥 ∈ ((⇝𝑡‘𝐽) “ (𝐴 ↑m ℕ)) ↔ ∃𝑓 ∈ (𝐴 ↑m ℕ)𝑓(⇝𝑡‘𝐽)𝑥)
19	16, 18	bitr4di 289	. . 3 ⊢ ((𝐽 ∈ Haus ∧ 𝐽 ∈ 1stω ∧ 𝐴 ⊆ 𝑋) → (𝑥 ∈ ((cls‘𝐽)‘𝐴) ↔ 𝑥 ∈ ((⇝𝑡‘𝐽) “ (𝐴 ↑m ℕ))))
20	19	eqrdv 2736	. 2 ⊢ ((𝐽 ∈ Haus ∧ 𝐽 ∈ 1stω ∧ 𝐴 ⊆ 𝑋) → ((cls‘𝐽)‘𝐴) = ((⇝𝑡‘𝐽) “ (𝐴 ↑m ℕ)))
21		ovex 7308	. . 3 ⊢ (𝐴 ↑m ℕ) ∈ V
22		lmfun 22532	. . . 4 ⊢ (𝐽 ∈ Haus → Fun (⇝𝑡‘𝐽))
23	22	3ad2ant1 1132	. . 3 ⊢ ((𝐽 ∈ Haus ∧ 𝐽 ∈ 1stω ∧ 𝐴 ⊆ 𝑋) → Fun (⇝𝑡‘𝐽))
24		imadomg 10290	. . 3 ⊢ ((𝐴 ↑m ℕ) ∈ V → (Fun (⇝𝑡‘𝐽) → ((⇝𝑡‘𝐽) “ (𝐴 ↑m ℕ)) ≼ (𝐴 ↑m ℕ)))
25	21, 23, 24	mpsyl 68	. 2 ⊢ ((𝐽 ∈ Haus ∧ 𝐽 ∈ 1stω ∧ 𝐴 ⊆ 𝑋) → ((⇝𝑡‘𝐽) “ (𝐴 ↑m ℕ)) ≼ (𝐴 ↑m ℕ))
26	20, 25	eqbrtrd 5096	1 ⊢ ((𝐽 ∈ Haus ∧ 𝐽 ∈ 1stω ∧ 𝐴 ⊆ 𝑋) → ((cls‘𝐽)‘𝐴) ≼ (𝐴 ↑m ℕ))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1086   = wceq 1539  ∃wex 1782   ∈ wcel 2106  ∃wrex 3065  Vcvv 3432   ⊆ wss 3887  ∪ cuni 4839   class class class wbr 5074   “ cima 5592  Fun wfun 6427  ⟶wf 6429  ‘cfv 6433  (class class class)co 7275   ↑_m cmap 8615   ≼ cdom 8731  ℕcn 11973  clsccl 22169  ⇝_𝑡clm 22377  Hauscha 22459  1^stωc1stc 22588

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  ax-inf2 9399  ax-cc 10191  ax-ac2 10219  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  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-nel 3050  df-ral 3069  df-rex 3070  df-rmo 3071  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-pss 3906  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-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-se 5545  df-we 5546  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-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  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-isom 6442  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-er 8498  df-map 8617  df-pm 8618  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-card 9697  df-acn 9700  df-ac 9872  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-nn 11974  df-n0 12234  df-z 12320  df-uz 12583  df-fz 13240  df-top 22043  df-topon 22060  df-cld 22170  df-ntr 22171  df-cls 22172  df-lm 22380  df-haus 22466  df-1stc 22590

This theorem is referenced by:  hauspwdom  22652