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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 22912 | . 2 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) → (𝑆 ∈ (Clsd‘𝐽) ↔ ∀𝑥∀𝑓((𝑓:ℕ⟶𝑆 ∧ 𝑓(⇝𝑡‘𝐽)𝑥) → 𝑥 ∈ 𝑆))) |
3 | 19.23v 1889 | . . . . 5 ⊢ (∀𝑓((𝑓:ℕ⟶𝑆 ∧ 𝑓(⇝𝑡‘𝐽)𝑥) → 𝑥 ∈ 𝑆) ↔ (∃𝑓(𝑓:ℕ⟶𝑆 ∧ 𝑓(⇝𝑡‘𝐽)𝑥) → 𝑥 ∈ 𝑆)) | |
4 | vex 3176 | . . . . . . . 8 ⊢ 𝑥 ∈ V | |
5 | 4 | elima2 5391 | . . . . . . 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 6130 | . . . . . . . . . . 11 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 ∈ dom ∞Met) | |
8 | ssexg 4732 | . . . . . . . . . . 11 ⊢ ((𝑆 ⊆ 𝑋 ∧ 𝑋 ∈ dom ∞Met) → 𝑆 ∈ V) | |
9 | 6, 7, 8 | syl2anr 494 | . . . . . . . . . 10 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) → 𝑆 ∈ V) |
10 | nnex 10903 | . . . . . . . . . 10 ⊢ ℕ ∈ V | |
11 | elmapg 7757 | . . . . . . . . . 10 ⊢ ((𝑆 ∈ V ∧ ℕ ∈ V) → (𝑓 ∈ (𝑆 ↑𝑚 ℕ) ↔ 𝑓:ℕ⟶𝑆)) | |
12 | 9, 10, 11 | sylancl 693 | . . . . . . . . 9 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) → (𝑓 ∈ (𝑆 ↑𝑚 ℕ) ↔ 𝑓:ℕ⟶𝑆)) |
13 | 12 | anbi1d 737 | . . . . . . . 8 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) → ((𝑓 ∈ (𝑆 ↑𝑚 ℕ) ∧ 𝑓(⇝𝑡‘𝐽)𝑥) ↔ (𝑓:ℕ⟶𝑆 ∧ 𝑓(⇝𝑡‘𝐽)𝑥))) |
14 | 13 | exbidv 1837 | . . . . . . 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 272 | . . . . . 6 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) → (∃𝑓(𝑓:ℕ⟶𝑆 ∧ 𝑓(⇝𝑡‘𝐽)𝑥) ↔ 𝑥 ∈ ((⇝𝑡‘𝐽) “ (𝑆 ↑𝑚 ℕ)))) |
16 | 15 | imbi1d 330 | . . . . 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 271 | . . . 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 1836 | . . 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 3557 | . . 3 ⊢ (((⇝𝑡‘𝐽) “ (𝑆 ↑𝑚 ℕ)) ⊆ 𝑆 ↔ ∀𝑥(𝑥 ∈ ((⇝𝑡‘𝐽) “ (𝑆 ↑𝑚 ℕ)) → 𝑥 ∈ 𝑆)) | |
20 | 18, 19 | syl6bbr 277 | . 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 267 | 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 195 ∧ wa 383 ∀wal 1473 = wceq 1475 ∃wex 1695 ∈ wcel 1977 Vcvv 3173 ⊆ wss 3540 class class class wbr 4583 dom cdm 5038 “ cima 5041 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ↑𝑚 cmap 7744 ℕcn 10897 ∞Metcxmt 19552 MetOpencmopn 19557 Clsdccld 20630 ⇝𝑡clm 20840 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cc 9140 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-pm 7747 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-inf 8232 df-card 8648 df-acn 8651 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-n0 11170 df-z 11255 df-uz 11564 df-q 11665 df-rp 11709 df-xneg 11822 df-xadd 11823 df-xmul 11824 df-fz 12198 df-topgen 15927 df-psmet 19559 df-xmet 19560 df-bl 19562 df-mopn 19563 df-top 20521 df-bases 20522 df-topon 20523 df-cld 20633 df-ntr 20634 df-cls 20635 df-lm 20843 df-1stc 21052 |
This theorem is referenced by: (None) |
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