![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > hausmapdom | Structured version Visualization version GIF version |
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 22013 to see a weaker result if the assumption of first-countability is omitted.) (Contributed by Mario Carneiro, 9-Apr-2015.) |
Ref | Expression |
---|---|
hauspwdom.1 | ⊢ 𝑋 = ∪ 𝐽 |
Ref | Expression |
---|---|
hausmapdom | ⊢ ((𝐽 ∈ Haus ∧ 𝐽 ∈ 1st𝜔 ∧ 𝐴 ⊆ 𝑋) → ((cls‘𝐽)‘𝐴) ≼ (𝐴 ↑𝑚 ℕ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hauspwdom.1 | . . . . . . . 8 ⊢ 𝑋 = ∪ 𝐽 | |
2 | 1 | 1stcelcls 21486 | . . . . . . 7 ⊢ ((𝐽 ∈ 1st𝜔 ∧ 𝐴 ⊆ 𝑋) → (𝑥 ∈ ((cls‘𝐽)‘𝐴) ↔ ∃𝑓(𝑓:ℕ⟶𝐴 ∧ 𝑓(⇝𝑡‘𝐽)𝑥))) |
3 | 2 | 3adant1 1125 | . . . . . 6 ⊢ ((𝐽 ∈ Haus ∧ 𝐽 ∈ 1st𝜔 ∧ 𝐴 ⊆ 𝑋) → (𝑥 ∈ ((cls‘𝐽)‘𝐴) ↔ ∃𝑓(𝑓:ℕ⟶𝐴 ∧ 𝑓(⇝𝑡‘𝐽)𝑥))) |
4 | uniexg 7121 | . . . . . . . . . . . 12 ⊢ (𝐽 ∈ Haus → ∪ 𝐽 ∈ V) | |
5 | 4 | 3ad2ant1 1128 | . . . . . . . . . . 11 ⊢ ((𝐽 ∈ Haus ∧ 𝐽 ∈ 1st𝜔 ∧ 𝐴 ⊆ 𝑋) → ∪ 𝐽 ∈ V) |
6 | 1, 5 | syl5eqel 2843 | . . . . . . . . . 10 ⊢ ((𝐽 ∈ Haus ∧ 𝐽 ∈ 1st𝜔 ∧ 𝐴 ⊆ 𝑋) → 𝑋 ∈ V) |
7 | simp3 1133 | . . . . . . . . . 10 ⊢ ((𝐽 ∈ Haus ∧ 𝐽 ∈ 1st𝜔 ∧ 𝐴 ⊆ 𝑋) → 𝐴 ⊆ 𝑋) | |
8 | 6, 7 | ssexd 4957 | . . . . . . . . 9 ⊢ ((𝐽 ∈ Haus ∧ 𝐽 ∈ 1st𝜔 ∧ 𝐴 ⊆ 𝑋) → 𝐴 ∈ V) |
9 | nnex 11238 | . . . . . . . . 9 ⊢ ℕ ∈ V | |
10 | elmapg 8038 | . . . . . . . . 9 ⊢ ((𝐴 ∈ V ∧ ℕ ∈ V) → (𝑓 ∈ (𝐴 ↑𝑚 ℕ) ↔ 𝑓:ℕ⟶𝐴)) | |
11 | 8, 9, 10 | sylancl 697 | . . . . . . . 8 ⊢ ((𝐽 ∈ Haus ∧ 𝐽 ∈ 1st𝜔 ∧ 𝐴 ⊆ 𝑋) → (𝑓 ∈ (𝐴 ↑𝑚 ℕ) ↔ 𝑓:ℕ⟶𝐴)) |
12 | 11 | anbi1d 743 | . . . . . . 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 1999 | . . . . . 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 271 | . . . . 5 ⊢ ((𝐽 ∈ Haus ∧ 𝐽 ∈ 1st𝜔 ∧ 𝐴 ⊆ 𝑋) → (𝑥 ∈ ((cls‘𝐽)‘𝐴) ↔ ∃𝑓(𝑓 ∈ (𝐴 ↑𝑚 ℕ) ∧ 𝑓(⇝𝑡‘𝐽)𝑥))) |
15 | df-rex 3056 | . . . . 5 ⊢ (∃𝑓 ∈ (𝐴 ↑𝑚 ℕ)𝑓(⇝𝑡‘𝐽)𝑥 ↔ ∃𝑓(𝑓 ∈ (𝐴 ↑𝑚 ℕ) ∧ 𝑓(⇝𝑡‘𝐽)𝑥)) | |
16 | 14, 15 | syl6bbr 278 | . . . 4 ⊢ ((𝐽 ∈ Haus ∧ 𝐽 ∈ 1st𝜔 ∧ 𝐴 ⊆ 𝑋) → (𝑥 ∈ ((cls‘𝐽)‘𝐴) ↔ ∃𝑓 ∈ (𝐴 ↑𝑚 ℕ)𝑓(⇝𝑡‘𝐽)𝑥)) |
17 | vex 3343 | . . . . 5 ⊢ 𝑥 ∈ V | |
18 | 17 | elima 5629 | . . . 4 ⊢ (𝑥 ∈ ((⇝𝑡‘𝐽) “ (𝐴 ↑𝑚 ℕ)) ↔ ∃𝑓 ∈ (𝐴 ↑𝑚 ℕ)𝑓(⇝𝑡‘𝐽)𝑥) |
19 | 16, 18 | syl6bbr 278 | . . 3 ⊢ ((𝐽 ∈ Haus ∧ 𝐽 ∈ 1st𝜔 ∧ 𝐴 ⊆ 𝑋) → (𝑥 ∈ ((cls‘𝐽)‘𝐴) ↔ 𝑥 ∈ ((⇝𝑡‘𝐽) “ (𝐴 ↑𝑚 ℕ)))) |
20 | 19 | eqrdv 2758 | . 2 ⊢ ((𝐽 ∈ Haus ∧ 𝐽 ∈ 1st𝜔 ∧ 𝐴 ⊆ 𝑋) → ((cls‘𝐽)‘𝐴) = ((⇝𝑡‘𝐽) “ (𝐴 ↑𝑚 ℕ))) |
21 | ovex 6842 | . . 3 ⊢ (𝐴 ↑𝑚 ℕ) ∈ V | |
22 | lmfun 21407 | . . . 4 ⊢ (𝐽 ∈ Haus → Fun (⇝𝑡‘𝐽)) | |
23 | 22 | 3ad2ant1 1128 | . . 3 ⊢ ((𝐽 ∈ Haus ∧ 𝐽 ∈ 1st𝜔 ∧ 𝐴 ⊆ 𝑋) → Fun (⇝𝑡‘𝐽)) |
24 | imadomg 9568 | . . 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 4826 | 1 ⊢ ((𝐽 ∈ Haus ∧ 𝐽 ∈ 1st𝜔 ∧ 𝐴 ⊆ 𝑋) → ((cls‘𝐽)‘𝐴) ≼ (𝐴 ↑𝑚 ℕ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 383 ∧ w3a 1072 = wceq 1632 ∃wex 1853 ∈ wcel 2139 ∃wrex 3051 Vcvv 3340 ⊆ wss 3715 ∪ cuni 4588 class class class wbr 4804 “ cima 5269 Fun wfun 6043 ⟶wf 6045 ‘cfv 6049 (class class class)co 6814 ↑𝑚 cmap 8025 ≼ cdom 8121 ℕcn 11232 clsccl 21044 ⇝𝑡clm 21252 Hauscha 21334 1st𝜔c1stc 21462 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-rep 4923 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7115 ax-inf2 8713 ax-cc 9469 ax-ac2 9497 ax-cnex 10204 ax-resscn 10205 ax-1cn 10206 ax-icn 10207 ax-addcl 10208 ax-addrcl 10209 ax-mulcl 10210 ax-mulrcl 10211 ax-mulcom 10212 ax-addass 10213 ax-mulass 10214 ax-distr 10215 ax-i2m1 10216 ax-1ne0 10217 ax-1rid 10218 ax-rnegex 10219 ax-rrecex 10220 ax-cnre 10221 ax-pre-lttri 10222 ax-pre-lttrn 10223 ax-pre-ltadd 10224 ax-pre-mulgt0 10225 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-nel 3036 df-ral 3055 df-rex 3056 df-reu 3057 df-rmo 3058 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-pss 3731 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-tp 4326 df-op 4328 df-uni 4589 df-int 4628 df-iun 4674 df-iin 4675 df-br 4805 df-opab 4865 df-mpt 4882 df-tr 4905 df-id 5174 df-eprel 5179 df-po 5187 df-so 5188 df-fr 5225 df-se 5226 df-we 5227 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-pred 5841 df-ord 5887 df-on 5888 df-lim 5889 df-suc 5890 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-isom 6058 df-riota 6775 df-ov 6817 df-oprab 6818 df-mpt2 6819 df-om 7232 df-1st 7334 df-2nd 7335 df-wrecs 7577 df-recs 7638 df-rdg 7676 df-1o 7730 df-oadd 7734 df-er 7913 df-map 8027 df-pm 8028 df-en 8124 df-dom 8125 df-sdom 8126 df-fin 8127 df-card 8975 df-acn 8978 df-ac 9149 df-pnf 10288 df-mnf 10289 df-xr 10290 df-ltxr 10291 df-le 10292 df-sub 10480 df-neg 10481 df-nn 11233 df-n0 11505 df-z 11590 df-uz 11900 df-fz 12540 df-top 20921 df-topon 20938 df-cld 21045 df-ntr 21046 df-cls 21047 df-lm 21255 df-haus 21341 df-1stc 21464 |
This theorem is referenced by: hauspwdom 21526 |
Copyright terms: Public domain | W3C validator |