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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 6696 | . 2 ⊢ (𝐷 ∈ (CMet‘𝑋) → 𝑋 ∈ V) | |
2 | elfvex 6696 | . . 3 ⊢ (𝐷 ∈ (Met‘𝑋) → 𝑋 ∈ V) | |
3 | 2 | adantr 481 | . 2 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ ∀𝑓 ∈ (CauFil‘𝐷)(𝐽 fLim 𝑓) ≠ ∅) → 𝑋 ∈ V) |
4 | fveq2 6663 | . . . . . 6 ⊢ (𝑥 = 𝑋 → (Met‘𝑥) = (Met‘𝑋)) | |
5 | 4 | rabeqdv 3482 | . . . . 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 23787 | . . . . 5 ⊢ CMet = (𝑥 ∈ V ↦ {𝑑 ∈ (Met‘𝑥) ∣ ∀𝑓 ∈ (CauFil‘𝑑)((MetOpen‘𝑑) fLim 𝑓) ≠ ∅}) | |
7 | fvex 6676 | . . . . . 6 ⊢ (Met‘𝑋) ∈ V | |
8 | 7 | rabex 5226 | . . . . 5 ⊢ {𝑑 ∈ (Met‘𝑋) ∣ ∀𝑓 ∈ (CauFil‘𝑑)((MetOpen‘𝑑) fLim 𝑓) ≠ ∅} ∈ V |
9 | 5, 6, 8 | fvmpt 6761 | . . . 4 ⊢ (𝑋 ∈ V → (CMet‘𝑋) = {𝑑 ∈ (Met‘𝑋) ∣ ∀𝑓 ∈ (CauFil‘𝑑)((MetOpen‘𝑑) fLim 𝑓) ≠ ∅}) |
10 | 9 | eleq2d 2895 | . . 3 ⊢ (𝑋 ∈ V → (𝐷 ∈ (CMet‘𝑋) ↔ 𝐷 ∈ {𝑑 ∈ (Met‘𝑋) ∣ ∀𝑓 ∈ (CauFil‘𝑑)((MetOpen‘𝑑) fLim 𝑓) ≠ ∅})) |
11 | fveq2 6663 | . . . . 5 ⊢ (𝑑 = 𝐷 → (CauFil‘𝑑) = (CauFil‘𝐷)) | |
12 | fveq2 6663 | . . . . . . . 8 ⊢ (𝑑 = 𝐷 → (MetOpen‘𝑑) = (MetOpen‘𝐷)) | |
13 | iscmet.1 | . . . . . . . 8 ⊢ 𝐽 = (MetOpen‘𝐷) | |
14 | 12, 13 | syl6eqr 2871 | . . . . . . 7 ⊢ (𝑑 = 𝐷 → (MetOpen‘𝑑) = 𝐽) |
15 | 14 | oveq1d 7160 | . . . . . 6 ⊢ (𝑑 = 𝐷 → ((MetOpen‘𝑑) fLim 𝑓) = (𝐽 fLim 𝑓)) |
16 | 15 | neeq1d 3072 | . . . . 5 ⊢ (𝑑 = 𝐷 → (((MetOpen‘𝑑) fLim 𝑓) ≠ ∅ ↔ (𝐽 fLim 𝑓) ≠ ∅)) |
17 | 11, 16 | raleqbidv 3399 | . . . 4 ⊢ (𝑑 = 𝐷 → (∀𝑓 ∈ (CauFil‘𝑑)((MetOpen‘𝑑) fLim 𝑓) ≠ ∅ ↔ ∀𝑓 ∈ (CauFil‘𝐷)(𝐽 fLim 𝑓) ≠ ∅)) |
18 | 17 | elrab 3677 | . . 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 288 | . 2 ⊢ (𝑋 ∈ V → (𝐷 ∈ (CMet‘𝑋) ↔ (𝐷 ∈ (Met‘𝑋) ∧ ∀𝑓 ∈ (CauFil‘𝐷)(𝐽 fLim 𝑓) ≠ ∅))) |
20 | 1, 3, 19 | pm5.21nii 380 | 1 ⊢ (𝐷 ∈ (CMet‘𝑋) ↔ (𝐷 ∈ (Met‘𝑋) ∧ ∀𝑓 ∈ (CauFil‘𝐷)(𝐽 fLim 𝑓) ≠ ∅)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ≠ wne 3013 ∀wral 3135 {crab 3139 Vcvv 3492 ∅c0 4288 ‘cfv 6348 (class class class)co 7145 Metcmet 20459 MetOpencmopn 20463 fLim cflim 22470 CauFilccfil 23782 CMetccmet 23784 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-iota 6307 df-fun 6350 df-fv 6356 df-ov 7148 df-cmet 23787 |
This theorem is referenced by: cmetcvg 23815 cmetmet 23816 iscmet3 23823 cmetss 23846 equivcmet 23847 relcmpcmet 23848 cmetcusp1 23883 |
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