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Mirrors > Home > MPE Home > Th. List > metelcls | Structured version Visualization version GIF version |
Description: A point belongs to the closure of a subset iff there is a sequence in the subset converging to it. Theorem 1.4-6(a) of [Kreyszig] p. 30. This proof uses countable choice ax-cc 9928. The statement can be generalized to first-countable spaces, not just metrizable spaces. (Contributed by NM, 8-Nov-2007.) (Proof shortened by Mario Carneiro, 1-May-2015.) |
Ref | Expression |
---|---|
metelcls.2 | ⊢ 𝐽 = (MetOpen‘𝐷) |
metelcls.3 | ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋)) |
metelcls.5 | ⊢ (𝜑 → 𝑆 ⊆ 𝑋) |
Ref | Expression |
---|---|
metelcls | ⊢ (𝜑 → (𝑃 ∈ ((cls‘𝐽)‘𝑆) ↔ ∃𝑓(𝑓:ℕ⟶𝑆 ∧ 𝑓(⇝𝑡‘𝐽)𝑃))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | metelcls.3 | . . 3 ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋)) | |
2 | metelcls.2 | . . . 4 ⊢ 𝐽 = (MetOpen‘𝐷) | |
3 | 2 | met1stc 23267 | . . 3 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ 1stω) |
4 | 1, 3 | syl 17 | . 2 ⊢ (𝜑 → 𝐽 ∈ 1stω) |
5 | metelcls.5 | . . 3 ⊢ (𝜑 → 𝑆 ⊆ 𝑋) | |
6 | 2 | mopnuni 23187 | . . . 4 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = ∪ 𝐽) |
7 | 1, 6 | syl 17 | . . 3 ⊢ (𝜑 → 𝑋 = ∪ 𝐽) |
8 | 5, 7 | sseqtrd 3915 | . 2 ⊢ (𝜑 → 𝑆 ⊆ ∪ 𝐽) |
9 | eqid 2738 | . . 3 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
10 | 9 | 1stcelcls 22205 | . 2 ⊢ ((𝐽 ∈ 1stω ∧ 𝑆 ⊆ ∪ 𝐽) → (𝑃 ∈ ((cls‘𝐽)‘𝑆) ↔ ∃𝑓(𝑓:ℕ⟶𝑆 ∧ 𝑓(⇝𝑡‘𝐽)𝑃))) |
11 | 4, 8, 10 | syl2anc 587 | 1 ⊢ (𝜑 → (𝑃 ∈ ((cls‘𝐽)‘𝑆) ↔ ∃𝑓(𝑓:ℕ⟶𝑆 ∧ 𝑓(⇝𝑡‘𝐽)𝑃))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1542 ∃wex 1786 ∈ wcel 2113 ⊆ wss 3841 ∪ cuni 4793 class class class wbr 5027 ⟶wf 6329 ‘cfv 6333 ℕcn 11709 ∞Metcxmet 20195 MetOpencmopn 20200 clsccl 21762 ⇝𝑡clm 21970 1stωc1stc 22181 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-rep 5151 ax-sep 5164 ax-nul 5171 ax-pow 5229 ax-pr 5293 ax-un 7473 ax-inf2 9170 ax-cc 9928 ax-cnex 10664 ax-resscn 10665 ax-1cn 10666 ax-icn 10667 ax-addcl 10668 ax-addrcl 10669 ax-mulcl 10670 ax-mulrcl 10671 ax-mulcom 10672 ax-addass 10673 ax-mulass 10674 ax-distr 10675 ax-i2m1 10676 ax-1ne0 10677 ax-1rid 10678 ax-rnegex 10679 ax-rrecex 10680 ax-cnre 10681 ax-pre-lttri 10682 ax-pre-lttrn 10683 ax-pre-ltadd 10684 ax-pre-mulgt0 10685 ax-pre-sup 10686 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3399 df-sbc 3680 df-csb 3789 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-pss 3860 df-nul 4210 df-if 4412 df-pw 4487 df-sn 4514 df-pr 4516 df-tp 4518 df-op 4520 df-uni 4794 df-int 4834 df-iun 4880 df-iin 4881 df-br 5028 df-opab 5090 df-mpt 5108 df-tr 5134 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6123 df-ord 6169 df-on 6170 df-lim 6171 df-suc 6172 df-iota 6291 df-fun 6335 df-fn 6336 df-f 6337 df-f1 6338 df-fo 6339 df-f1o 6340 df-fv 6341 df-isom 6342 df-riota 7121 df-ov 7167 df-oprab 7168 df-mpo 7169 df-om 7594 df-1st 7707 df-2nd 7708 df-wrecs 7969 df-recs 8030 df-rdg 8068 df-1o 8124 df-er 8313 df-map 8432 df-pm 8433 df-en 8549 df-dom 8550 df-sdom 8551 df-fin 8552 df-sup 8972 df-inf 8973 df-card 9434 df-acn 9437 df-pnf 10748 df-mnf 10749 df-xr 10750 df-ltxr 10751 df-le 10752 df-sub 10943 df-neg 10944 df-div 11369 df-nn 11710 df-2 11772 df-n0 11970 df-z 12056 df-uz 12318 df-q 12424 df-rp 12466 df-xneg 12583 df-xadd 12584 df-xmul 12585 df-fz 12975 df-topgen 16813 df-psmet 20202 df-xmet 20203 df-bl 20205 df-mopn 20206 df-top 21638 df-topon 21655 df-bases 21690 df-cld 21763 df-ntr 21764 df-cls 21765 df-lm 21973 df-1stc 22183 |
This theorem is referenced by: metcld 24051 |
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