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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 2736 | . . . 4 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
2 | 1 | 2ndcsep 22310 | . . 3 ⊢ (𝐽 ∈ 2ndω → ∃𝑥 ∈ 𝒫 ∪ 𝐽(𝑥 ≼ ω ∧ ((cls‘𝐽)‘𝑥) = ∪ 𝐽)) |
3 | methaus.1 | . . . . . 6 ⊢ 𝐽 = (MetOpen‘𝐷) | |
4 | 3 | mopnuni 23293 | . . . . 5 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = ∪ 𝐽) |
5 | 4 | pweqd 4518 | . . . 4 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝒫 𝑋 = 𝒫 ∪ 𝐽) |
6 | 4 | eqeq2d 2747 | . . . . 5 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (((cls‘𝐽)‘𝑥) = 𝑋 ↔ ((cls‘𝐽)‘𝑥) = ∪ 𝐽)) |
7 | 6 | anbi2d 632 | . . . 4 ⊢ (𝐷 ∈ (∞Met‘𝑋) → ((𝑥 ≼ ω ∧ ((cls‘𝐽)‘𝑥) = 𝑋) ↔ (𝑥 ≼ ω ∧ ((cls‘𝐽)‘𝑥) = ∪ 𝐽))) |
8 | 5, 7 | rexeqbidv 3304 | . . 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 249 | . 2 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝐽 ∈ 2ndω → ∃𝑥 ∈ 𝒫 𝑋(𝑥 ≼ ω ∧ ((cls‘𝐽)‘𝑥) = 𝑋))) |
10 | elpwi 4508 | . . . 4 ⊢ (𝑥 ∈ 𝒫 𝑋 → 𝑥 ⊆ 𝑋) | |
11 | 3 | met2ndci 23374 | . . . . . 6 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑥 ⊆ 𝑋 ∧ 𝑥 ≼ ω ∧ ((cls‘𝐽)‘𝑥) = 𝑋)) → 𝐽 ∈ 2ndω) |
12 | 11 | 3exp2 1356 | . . . . 5 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝑥 ⊆ 𝑋 → (𝑥 ≼ ω → (((cls‘𝐽)‘𝑥) = 𝑋 → 𝐽 ∈ 2ndω)))) |
13 | 12 | imp4a 426 | . . . 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 3192 | . 2 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (∃𝑥 ∈ 𝒫 𝑋(𝑥 ≼ ω ∧ ((cls‘𝐽)‘𝑥) = 𝑋) → 𝐽 ∈ 2ndω)) |
16 | 9, 15 | impbid 215 | 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 209 ∧ wa 399 = wceq 1543 ∈ wcel 2112 ∃wrex 3052 ⊆ wss 3853 𝒫 cpw 4499 ∪ cuni 4805 class class class wbr 5039 ‘cfv 6358 ωcom 7622 ≼ cdom 8602 ∞Metcxmet 20302 MetOpencmopn 20307 clsccl 21869 2ndωc2ndc 22289 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-inf2 9234 ax-cc 10014 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 ax-pre-sup 10772 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-int 4846 df-iun 4892 df-iin 4893 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-se 5495 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-isom 6367 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-1st 7739 df-2nd 7740 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-1o 8180 df-er 8369 df-map 8488 df-en 8605 df-dom 8606 df-sdom 8607 df-fin 8608 df-sup 9036 df-inf 9037 df-oi 9104 df-card 9520 df-acn 9523 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-div 11455 df-nn 11796 df-2 11858 df-n0 12056 df-z 12142 df-uz 12404 df-q 12510 df-rp 12552 df-xneg 12669 df-xadd 12670 df-xmul 12671 df-topgen 16902 df-psmet 20309 df-xmet 20310 df-bl 20312 df-mopn 20313 df-top 21745 df-topon 21762 df-bases 21797 df-cld 21870 df-ntr 21871 df-cls 21872 df-2ndc 22291 |
This theorem is referenced by: (None) |
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