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Mirrors > Home > MPE Home > Th. List > metcld2 | Structured version Visualization version GIF version |
Description: A subset of a metric space is closed iff every convergent sequence on it converges to a point in the subset. Theorem 1.4-6(b) of [Kreyszig] p. 30. (Contributed by Mario Carneiro, 1-May-2014.) |
Ref | Expression |
---|---|
metcld.2 | ⊢ 𝐽 = (MetOpen‘𝐷) |
Ref | Expression |
---|---|
metcld2 | ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) → (𝑆 ∈ (Clsd‘𝐽) ↔ ((⇝𝑡‘𝐽) “ (𝑆 ↑m ℕ)) ⊆ 𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | metcld.2 | . . 3 ⊢ 𝐽 = (MetOpen‘𝐷) | |
2 | 1 | metcld 23911 | . 2 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) → (𝑆 ∈ (Clsd‘𝐽) ↔ ∀𝑥∀𝑓((𝑓:ℕ⟶𝑆 ∧ 𝑓(⇝𝑡‘𝐽)𝑥) → 𝑥 ∈ 𝑆))) |
3 | 19.23v 1943 | . . . . 5 ⊢ (∀𝑓((𝑓:ℕ⟶𝑆 ∧ 𝑓(⇝𝑡‘𝐽)𝑥) → 𝑥 ∈ 𝑆) ↔ (∃𝑓(𝑓:ℕ⟶𝑆 ∧ 𝑓(⇝𝑡‘𝐽)𝑥) → 𝑥 ∈ 𝑆)) | |
4 | vex 3499 | . . . . . . . 8 ⊢ 𝑥 ∈ V | |
5 | 4 | elima2 5937 | . . . . . . 7 ⊢ (𝑥 ∈ ((⇝𝑡‘𝐽) “ (𝑆 ↑m ℕ)) ↔ ∃𝑓(𝑓 ∈ (𝑆 ↑m ℕ) ∧ 𝑓(⇝𝑡‘𝐽)𝑥)) |
6 | id 22 | . . . . . . . . . . 11 ⊢ (𝑆 ⊆ 𝑋 → 𝑆 ⊆ 𝑋) | |
7 | elfvdm 6704 | . . . . . . . . . . 11 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 ∈ dom ∞Met) | |
8 | ssexg 5229 | . . . . . . . . . . 11 ⊢ ((𝑆 ⊆ 𝑋 ∧ 𝑋 ∈ dom ∞Met) → 𝑆 ∈ V) | |
9 | 6, 7, 8 | syl2anr 598 | . . . . . . . . . 10 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) → 𝑆 ∈ V) |
10 | nnex 11646 | . . . . . . . . . 10 ⊢ ℕ ∈ V | |
11 | elmapg 8421 | . . . . . . . . . 10 ⊢ ((𝑆 ∈ V ∧ ℕ ∈ V) → (𝑓 ∈ (𝑆 ↑m ℕ) ↔ 𝑓:ℕ⟶𝑆)) | |
12 | 9, 10, 11 | sylancl 588 | . . . . . . . . 9 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) → (𝑓 ∈ (𝑆 ↑m ℕ) ↔ 𝑓:ℕ⟶𝑆)) |
13 | 12 | anbi1d 631 | . . . . . . . 8 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) → ((𝑓 ∈ (𝑆 ↑m ℕ) ∧ 𝑓(⇝𝑡‘𝐽)𝑥) ↔ (𝑓:ℕ⟶𝑆 ∧ 𝑓(⇝𝑡‘𝐽)𝑥))) |
14 | 13 | exbidv 1922 | . . . . . . 7 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) → (∃𝑓(𝑓 ∈ (𝑆 ↑m ℕ) ∧ 𝑓(⇝𝑡‘𝐽)𝑥) ↔ ∃𝑓(𝑓:ℕ⟶𝑆 ∧ 𝑓(⇝𝑡‘𝐽)𝑥))) |
15 | 5, 14 | syl5rbb 286 | . . . . . 6 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) → (∃𝑓(𝑓:ℕ⟶𝑆 ∧ 𝑓(⇝𝑡‘𝐽)𝑥) ↔ 𝑥 ∈ ((⇝𝑡‘𝐽) “ (𝑆 ↑m ℕ)))) |
16 | 15 | imbi1d 344 | . . . . 5 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) → ((∃𝑓(𝑓:ℕ⟶𝑆 ∧ 𝑓(⇝𝑡‘𝐽)𝑥) → 𝑥 ∈ 𝑆) ↔ (𝑥 ∈ ((⇝𝑡‘𝐽) “ (𝑆 ↑m ℕ)) → 𝑥 ∈ 𝑆))) |
17 | 3, 16 | syl5bb 285 | . . . 4 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) → (∀𝑓((𝑓:ℕ⟶𝑆 ∧ 𝑓(⇝𝑡‘𝐽)𝑥) → 𝑥 ∈ 𝑆) ↔ (𝑥 ∈ ((⇝𝑡‘𝐽) “ (𝑆 ↑m ℕ)) → 𝑥 ∈ 𝑆))) |
18 | 17 | albidv 1921 | . . 3 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) → (∀𝑥∀𝑓((𝑓:ℕ⟶𝑆 ∧ 𝑓(⇝𝑡‘𝐽)𝑥) → 𝑥 ∈ 𝑆) ↔ ∀𝑥(𝑥 ∈ ((⇝𝑡‘𝐽) “ (𝑆 ↑m ℕ)) → 𝑥 ∈ 𝑆))) |
19 | dfss2 3957 | . . 3 ⊢ (((⇝𝑡‘𝐽) “ (𝑆 ↑m ℕ)) ⊆ 𝑆 ↔ ∀𝑥(𝑥 ∈ ((⇝𝑡‘𝐽) “ (𝑆 ↑m ℕ)) → 𝑥 ∈ 𝑆)) | |
20 | 18, 19 | syl6bbr 291 | . 2 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) → (∀𝑥∀𝑓((𝑓:ℕ⟶𝑆 ∧ 𝑓(⇝𝑡‘𝐽)𝑥) → 𝑥 ∈ 𝑆) ↔ ((⇝𝑡‘𝐽) “ (𝑆 ↑m ℕ)) ⊆ 𝑆)) |
21 | 2, 20 | bitrd 281 | 1 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) → (𝑆 ∈ (Clsd‘𝐽) ↔ ((⇝𝑡‘𝐽) “ (𝑆 ↑m ℕ)) ⊆ 𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∀wal 1535 = wceq 1537 ∃wex 1780 ∈ wcel 2114 Vcvv 3496 ⊆ wss 3938 class class class wbr 5068 dom cdm 5557 “ cima 5560 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 ↑m cmap 8408 ℕcn 11640 ∞Metcxmet 20532 MetOpencmopn 20537 Clsdccld 21626 ⇝𝑡clm 21836 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-inf2 9106 ax-cc 9859 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-iin 4924 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-map 8410 df-pm 8411 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-sup 8908 df-inf 8909 df-card 9370 df-acn 9373 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-n0 11901 df-z 11985 df-uz 12247 df-q 12352 df-rp 12393 df-xneg 12510 df-xadd 12511 df-xmul 12512 df-fz 12896 df-topgen 16719 df-psmet 20539 df-xmet 20540 df-bl 20542 df-mopn 20543 df-top 21504 df-topon 21521 df-bases 21556 df-cld 21629 df-ntr 21630 df-cls 21631 df-lm 21839 df-1stc 22049 |
This theorem is referenced by: (None) |
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