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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 2820 | . . . 4 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
2 | 1 | 2ndcsep 22062 | . . 3 ⊢ (𝐽 ∈ 2ndω → ∃𝑥 ∈ 𝒫 ∪ 𝐽(𝑥 ≼ ω ∧ ((cls‘𝐽)‘𝑥) = ∪ 𝐽)) |
3 | methaus.1 | . . . . . 6 ⊢ 𝐽 = (MetOpen‘𝐷) | |
4 | 3 | mopnuni 23046 | . . . . 5 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = ∪ 𝐽) |
5 | 4 | pweqd 4551 | . . . 4 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝒫 𝑋 = 𝒫 ∪ 𝐽) |
6 | 4 | eqeq2d 2831 | . . . . 5 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (((cls‘𝐽)‘𝑥) = 𝑋 ↔ ((cls‘𝐽)‘𝑥) = ∪ 𝐽)) |
7 | 6 | anbi2d 630 | . . . 4 ⊢ (𝐷 ∈ (∞Met‘𝑋) → ((𝑥 ≼ ω ∧ ((cls‘𝐽)‘𝑥) = 𝑋) ↔ (𝑥 ≼ ω ∧ ((cls‘𝐽)‘𝑥) = ∪ 𝐽))) |
8 | 5, 7 | rexeqbidv 3401 | . . 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 248 | . 2 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝐽 ∈ 2ndω → ∃𝑥 ∈ 𝒫 𝑋(𝑥 ≼ ω ∧ ((cls‘𝐽)‘𝑥) = 𝑋))) |
10 | elpwi 4541 | . . . 4 ⊢ (𝑥 ∈ 𝒫 𝑋 → 𝑥 ⊆ 𝑋) | |
11 | 3 | met2ndci 23127 | . . . . . 6 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑥 ⊆ 𝑋 ∧ 𝑥 ≼ ω ∧ ((cls‘𝐽)‘𝑥) = 𝑋)) → 𝐽 ∈ 2ndω) |
12 | 11 | 3exp2 1349 | . . . . 5 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝑥 ⊆ 𝑋 → (𝑥 ≼ ω → (((cls‘𝐽)‘𝑥) = 𝑋 → 𝐽 ∈ 2ndω)))) |
13 | 12 | imp4a 425 | . . . 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 3282 | . 2 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (∃𝑥 ∈ 𝒫 𝑋(𝑥 ≼ ω ∧ ((cls‘𝐽)‘𝑥) = 𝑋) → 𝐽 ∈ 2ndω)) |
16 | 9, 15 | impbid 214 | 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 208 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ∃wrex 3138 ⊆ wss 3929 𝒫 cpw 4532 ∪ cuni 4831 class class class wbr 5059 ‘cfv 6348 ωcom 7573 ≼ cdom 8500 ∞Metcxmet 20525 MetOpencmopn 20530 clsccl 21621 2ndωc2ndc 22041 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 ax-inf2 9097 ax-cc 9850 ax-cnex 10586 ax-resscn 10587 ax-1cn 10588 ax-icn 10589 ax-addcl 10590 ax-addrcl 10591 ax-mulcl 10592 ax-mulrcl 10593 ax-mulcom 10594 ax-addass 10595 ax-mulass 10596 ax-distr 10597 ax-i2m1 10598 ax-1ne0 10599 ax-1rid 10600 ax-rnegex 10601 ax-rrecex 10602 ax-cnre 10603 ax-pre-lttri 10604 ax-pre-lttrn 10605 ax-pre-ltadd 10606 ax-pre-mulgt0 10607 ax-pre-sup 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-nel 3123 df-ral 3142 df-rex 3143 df-reu 3144 df-rmo 3145 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-pss 3947 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4870 df-iun 4914 df-iin 4915 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7107 df-ov 7152 df-oprab 7153 df-mpo 7154 df-om 7574 df-1st 7682 df-2nd 7683 df-wrecs 7940 df-recs 8001 df-rdg 8039 df-1o 8095 df-oadd 8099 df-er 8282 df-map 8401 df-en 8503 df-dom 8504 df-sdom 8505 df-fin 8506 df-sup 8899 df-inf 8900 df-oi 8967 df-card 9361 df-acn 9364 df-pnf 10670 df-mnf 10671 df-xr 10672 df-ltxr 10673 df-le 10674 df-sub 10865 df-neg 10866 df-div 11291 df-nn 11632 df-2 11694 df-n0 11892 df-z 11976 df-uz 12238 df-q 12343 df-rp 12384 df-xneg 12501 df-xadd 12502 df-xmul 12503 df-topgen 16712 df-psmet 20532 df-xmet 20533 df-bl 20535 df-mopn 20536 df-top 21497 df-topon 21514 df-bases 21549 df-cld 21622 df-ntr 21623 df-cls 21624 df-2ndc 22043 |
This theorem is referenced by: (None) |
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