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Mirrors > Home > MPE Home > Th. List > iscmet | Structured version Visualization version GIF version |
Description: The property "𝐷 is a complete metric." meaning all Cauchy filters converge to a point in the space. (Contributed by Mario Carneiro, 1-May-2014.) (Revised by Mario Carneiro, 13-Oct-2015.) |
Ref | Expression |
---|---|
iscmet.1 | ⊢ 𝐽 = (MetOpen‘𝐷) |
Ref | Expression |
---|---|
iscmet | ⊢ (𝐷 ∈ (CMet‘𝑋) ↔ (𝐷 ∈ (Met‘𝑋) ∧ ∀𝑓 ∈ (CauFil‘𝐷)(𝐽 fLim 𝑓) ≠ ∅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfvex 6697 | . 2 ⊢ (𝐷 ∈ (CMet‘𝑋) → 𝑋 ∈ V) | |
2 | elfvex 6697 | . . 3 ⊢ (𝐷 ∈ (Met‘𝑋) → 𝑋 ∈ V) | |
3 | 2 | adantr 483 | . 2 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ ∀𝑓 ∈ (CauFil‘𝐷)(𝐽 fLim 𝑓) ≠ ∅) → 𝑋 ∈ V) |
4 | fveq2 6664 | . . . . . 6 ⊢ (𝑥 = 𝑋 → (Met‘𝑥) = (Met‘𝑋)) | |
5 | 4 | rabeqdv 3484 | . . . . 5 ⊢ (𝑥 = 𝑋 → {𝑑 ∈ (Met‘𝑥) ∣ ∀𝑓 ∈ (CauFil‘𝑑)((MetOpen‘𝑑) fLim 𝑓) ≠ ∅} = {𝑑 ∈ (Met‘𝑋) ∣ ∀𝑓 ∈ (CauFil‘𝑑)((MetOpen‘𝑑) fLim 𝑓) ≠ ∅}) |
6 | df-cmet 23854 | . . . . 5 ⊢ CMet = (𝑥 ∈ V ↦ {𝑑 ∈ (Met‘𝑥) ∣ ∀𝑓 ∈ (CauFil‘𝑑)((MetOpen‘𝑑) fLim 𝑓) ≠ ∅}) | |
7 | fvex 6677 | . . . . . 6 ⊢ (Met‘𝑋) ∈ V | |
8 | 7 | rabex 5227 | . . . . 5 ⊢ {𝑑 ∈ (Met‘𝑋) ∣ ∀𝑓 ∈ (CauFil‘𝑑)((MetOpen‘𝑑) fLim 𝑓) ≠ ∅} ∈ V |
9 | 5, 6, 8 | fvmpt 6762 | . . . 4 ⊢ (𝑋 ∈ V → (CMet‘𝑋) = {𝑑 ∈ (Met‘𝑋) ∣ ∀𝑓 ∈ (CauFil‘𝑑)((MetOpen‘𝑑) fLim 𝑓) ≠ ∅}) |
10 | 9 | eleq2d 2898 | . . 3 ⊢ (𝑋 ∈ V → (𝐷 ∈ (CMet‘𝑋) ↔ 𝐷 ∈ {𝑑 ∈ (Met‘𝑋) ∣ ∀𝑓 ∈ (CauFil‘𝑑)((MetOpen‘𝑑) fLim 𝑓) ≠ ∅})) |
11 | fveq2 6664 | . . . . 5 ⊢ (𝑑 = 𝐷 → (CauFil‘𝑑) = (CauFil‘𝐷)) | |
12 | fveq2 6664 | . . . . . . . 8 ⊢ (𝑑 = 𝐷 → (MetOpen‘𝑑) = (MetOpen‘𝐷)) | |
13 | iscmet.1 | . . . . . . . 8 ⊢ 𝐽 = (MetOpen‘𝐷) | |
14 | 12, 13 | syl6eqr 2874 | . . . . . . 7 ⊢ (𝑑 = 𝐷 → (MetOpen‘𝑑) = 𝐽) |
15 | 14 | oveq1d 7165 | . . . . . 6 ⊢ (𝑑 = 𝐷 → ((MetOpen‘𝑑) fLim 𝑓) = (𝐽 fLim 𝑓)) |
16 | 15 | neeq1d 3075 | . . . . 5 ⊢ (𝑑 = 𝐷 → (((MetOpen‘𝑑) fLim 𝑓) ≠ ∅ ↔ (𝐽 fLim 𝑓) ≠ ∅)) |
17 | 11, 16 | raleqbidv 3401 | . . . 4 ⊢ (𝑑 = 𝐷 → (∀𝑓 ∈ (CauFil‘𝑑)((MetOpen‘𝑑) fLim 𝑓) ≠ ∅ ↔ ∀𝑓 ∈ (CauFil‘𝐷)(𝐽 fLim 𝑓) ≠ ∅)) |
18 | 17 | elrab 3679 | . . 3 ⊢ (𝐷 ∈ {𝑑 ∈ (Met‘𝑋) ∣ ∀𝑓 ∈ (CauFil‘𝑑)((MetOpen‘𝑑) fLim 𝑓) ≠ ∅} ↔ (𝐷 ∈ (Met‘𝑋) ∧ ∀𝑓 ∈ (CauFil‘𝐷)(𝐽 fLim 𝑓) ≠ ∅)) |
19 | 10, 18 | syl6bb 289 | . 2 ⊢ (𝑋 ∈ V → (𝐷 ∈ (CMet‘𝑋) ↔ (𝐷 ∈ (Met‘𝑋) ∧ ∀𝑓 ∈ (CauFil‘𝐷)(𝐽 fLim 𝑓) ≠ ∅))) |
20 | 1, 3, 19 | pm5.21nii 382 | 1 ⊢ (𝐷 ∈ (CMet‘𝑋) ↔ (𝐷 ∈ (Met‘𝑋) ∧ ∀𝑓 ∈ (CauFil‘𝐷)(𝐽 fLim 𝑓) ≠ ∅)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 ∀wral 3138 {crab 3142 Vcvv 3494 ∅c0 4290 ‘cfv 6349 (class class class)co 7150 Metcmet 20525 MetOpencmopn 20529 fLim cflim 22536 CauFilccfil 23849 CMetccmet 23851 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-iota 6308 df-fun 6351 df-fv 6357 df-ov 7153 df-cmet 23854 |
This theorem is referenced by: cmetcvg 23882 cmetmet 23883 iscmet3 23890 cmetss 23913 equivcmet 23914 relcmpcmet 23915 cmetcusp1 23950 |
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