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Mirrors > Home > MPE Home > Th. List > met2ndc | Structured version Visualization version GIF version |
Description: A metric space is second-countable iff it is separable (has a countable dense subset). (Contributed by Mario Carneiro, 13-Apr-2015.) |
Ref | Expression |
---|---|
methaus.1 | ⊢ 𝐽 = (MetOpen‘𝐷) |
Ref | Expression |
---|---|
met2ndc | ⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝐽 ∈ 2nd𝜔 ↔ ∃𝑥 ∈ 𝒫 𝑋(𝑥 ≼ ω ∧ ((cls‘𝐽)‘𝑥) = 𝑋))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2778 | . . . 4 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
2 | 1 | 2ndcsep 21771 | . . 3 ⊢ (𝐽 ∈ 2nd𝜔 → ∃𝑥 ∈ 𝒫 ∪ 𝐽(𝑥 ≼ ω ∧ ((cls‘𝐽)‘𝑥) = ∪ 𝐽)) |
3 | methaus.1 | . . . . . 6 ⊢ 𝐽 = (MetOpen‘𝐷) | |
4 | 3 | mopnuni 22754 | . . . . 5 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = ∪ 𝐽) |
5 | 4 | pweqd 4427 | . . . 4 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝒫 𝑋 = 𝒫 ∪ 𝐽) |
6 | 4 | eqeq2d 2788 | . . . . 5 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (((cls‘𝐽)‘𝑥) = 𝑋 ↔ ((cls‘𝐽)‘𝑥) = ∪ 𝐽)) |
7 | 6 | anbi2d 619 | . . . 4 ⊢ (𝐷 ∈ (∞Met‘𝑋) → ((𝑥 ≼ ω ∧ ((cls‘𝐽)‘𝑥) = 𝑋) ↔ (𝑥 ≼ ω ∧ ((cls‘𝐽)‘𝑥) = ∪ 𝐽))) |
8 | 5, 7 | rexeqbidv 3342 | . . 3 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (∃𝑥 ∈ 𝒫 𝑋(𝑥 ≼ ω ∧ ((cls‘𝐽)‘𝑥) = 𝑋) ↔ ∃𝑥 ∈ 𝒫 ∪ 𝐽(𝑥 ≼ ω ∧ ((cls‘𝐽)‘𝑥) = ∪ 𝐽))) |
9 | 2, 8 | syl5ibr 238 | . 2 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝐽 ∈ 2nd𝜔 → ∃𝑥 ∈ 𝒫 𝑋(𝑥 ≼ ω ∧ ((cls‘𝐽)‘𝑥) = 𝑋))) |
10 | elpwi 4432 | . . . 4 ⊢ (𝑥 ∈ 𝒫 𝑋 → 𝑥 ⊆ 𝑋) | |
11 | 3 | met2ndci 22835 | . . . . . 6 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑥 ⊆ 𝑋 ∧ 𝑥 ≼ ω ∧ ((cls‘𝐽)‘𝑥) = 𝑋)) → 𝐽 ∈ 2nd𝜔) |
12 | 11 | 3exp2 1334 | . . . . 5 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝑥 ⊆ 𝑋 → (𝑥 ≼ ω → (((cls‘𝐽)‘𝑥) = 𝑋 → 𝐽 ∈ 2nd𝜔)))) |
13 | 12 | imp4a 415 | . . . 4 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝑥 ⊆ 𝑋 → ((𝑥 ≼ ω ∧ ((cls‘𝐽)‘𝑥) = 𝑋) → 𝐽 ∈ 2nd𝜔))) |
14 | 10, 13 | syl5 34 | . . 3 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝑥 ∈ 𝒫 𝑋 → ((𝑥 ≼ ω ∧ ((cls‘𝐽)‘𝑥) = 𝑋) → 𝐽 ∈ 2nd𝜔))) |
15 | 14 | rexlimdv 3228 | . 2 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (∃𝑥 ∈ 𝒫 𝑋(𝑥 ≼ ω ∧ ((cls‘𝐽)‘𝑥) = 𝑋) → 𝐽 ∈ 2nd𝜔)) |
16 | 9, 15 | impbid 204 | 1 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝐽 ∈ 2nd𝜔 ↔ ∃𝑥 ∈ 𝒫 𝑋(𝑥 ≼ ω ∧ ((cls‘𝐽)‘𝑥) = 𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 387 = wceq 1507 ∈ wcel 2050 ∃wrex 3089 ⊆ wss 3829 𝒫 cpw 4422 ∪ cuni 4712 class class class wbr 4929 ‘cfv 6188 ωcom 7396 ≼ cdom 8304 ∞Metcxmet 20232 MetOpencmopn 20237 clsccl 21330 2nd𝜔c2ndc 21750 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-rep 5049 ax-sep 5060 ax-nul 5067 ax-pow 5119 ax-pr 5186 ax-un 7279 ax-inf2 8898 ax-cc 9655 ax-cnex 10391 ax-resscn 10392 ax-1cn 10393 ax-icn 10394 ax-addcl 10395 ax-addrcl 10396 ax-mulcl 10397 ax-mulrcl 10398 ax-mulcom 10399 ax-addass 10400 ax-mulass 10401 ax-distr 10402 ax-i2m1 10403 ax-1ne0 10404 ax-1rid 10405 ax-rnegex 10406 ax-rrecex 10407 ax-cnre 10408 ax-pre-lttri 10409 ax-pre-lttrn 10410 ax-pre-ltadd 10411 ax-pre-mulgt0 10412 ax-pre-sup 10413 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ne 2968 df-nel 3074 df-ral 3093 df-rex 3094 df-reu 3095 df-rmo 3096 df-rab 3097 df-v 3417 df-sbc 3682 df-csb 3787 df-dif 3832 df-un 3834 df-in 3836 df-ss 3843 df-pss 3845 df-nul 4179 df-if 4351 df-pw 4424 df-sn 4442 df-pr 4444 df-tp 4446 df-op 4448 df-uni 4713 df-int 4750 df-iun 4794 df-iin 4795 df-br 4930 df-opab 4992 df-mpt 5009 df-tr 5031 df-id 5312 df-eprel 5317 df-po 5326 df-so 5327 df-fr 5366 df-se 5367 df-we 5368 df-xp 5413 df-rel 5414 df-cnv 5415 df-co 5416 df-dm 5417 df-rn 5418 df-res 5419 df-ima 5420 df-pred 5986 df-ord 6032 df-on 6033 df-lim 6034 df-suc 6035 df-iota 6152 df-fun 6190 df-fn 6191 df-f 6192 df-f1 6193 df-fo 6194 df-f1o 6195 df-fv 6196 df-isom 6197 df-riota 6937 df-ov 6979 df-oprab 6980 df-mpo 6981 df-om 7397 df-1st 7501 df-2nd 7502 df-wrecs 7750 df-recs 7812 df-rdg 7850 df-1o 7905 df-oadd 7909 df-er 8089 df-map 8208 df-en 8307 df-dom 8308 df-sdom 8309 df-fin 8310 df-sup 8701 df-inf 8702 df-oi 8769 df-card 9162 df-acn 9165 df-pnf 10476 df-mnf 10477 df-xr 10478 df-ltxr 10479 df-le 10480 df-sub 10672 df-neg 10673 df-div 11099 df-nn 11440 df-2 11503 df-n0 11708 df-z 11794 df-uz 12059 df-q 12163 df-rp 12205 df-xneg 12324 df-xadd 12325 df-xmul 12326 df-topgen 16573 df-psmet 20239 df-xmet 20240 df-bl 20242 df-mopn 20243 df-top 21206 df-topon 21223 df-bases 21258 df-cld 21331 df-ntr 21332 df-cls 21333 df-2ndc 21752 |
This theorem is referenced by: (None) |
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