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Mirrors > Home > MPE Home > Th. List > xmetunirn | Structured version Visualization version GIF version |
Description: Two ways to express an extended metric on an unspecified base. (Contributed by Mario Carneiro, 13-Oct-2015.) |
Ref | Expression |
---|---|
xmetunirn | ⊢ (𝐷 ∈ ∪ ran ∞Met ↔ 𝐷 ∈ (∞Met‘dom dom 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ovex 6839 | . . . . . 6 ⊢ (ℝ* ↑𝑚 (𝑥 × 𝑥)) ∈ V | |
2 | 1 | rabex 4962 | . . . . 5 ⊢ {𝑑 ∈ (ℝ* ↑𝑚 (𝑥 × 𝑥)) ∣ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (((𝑦𝑑𝑧) = 0 ↔ 𝑦 = 𝑧) ∧ ∀𝑤 ∈ 𝑥 (𝑦𝑑𝑧) ≤ ((𝑤𝑑𝑦) +𝑒 (𝑤𝑑𝑧)))} ∈ V |
3 | df-xmet 19939 | . . . . 5 ⊢ ∞Met = (𝑥 ∈ V ↦ {𝑑 ∈ (ℝ* ↑𝑚 (𝑥 × 𝑥)) ∣ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (((𝑦𝑑𝑧) = 0 ↔ 𝑦 = 𝑧) ∧ ∀𝑤 ∈ 𝑥 (𝑦𝑑𝑧) ≤ ((𝑤𝑑𝑦) +𝑒 (𝑤𝑑𝑧)))}) | |
4 | 2, 3 | fnmpti 6181 | . . . 4 ⊢ ∞Met Fn V |
5 | fnunirn 6672 | . . . 4 ⊢ (∞Met Fn V → (𝐷 ∈ ∪ ran ∞Met ↔ ∃𝑥 ∈ V 𝐷 ∈ (∞Met‘𝑥))) | |
6 | 4, 5 | ax-mp 5 | . . 3 ⊢ (𝐷 ∈ ∪ ran ∞Met ↔ ∃𝑥 ∈ V 𝐷 ∈ (∞Met‘𝑥)) |
7 | id 22 | . . . . 5 ⊢ (𝐷 ∈ (∞Met‘𝑥) → 𝐷 ∈ (∞Met‘𝑥)) | |
8 | xmetdmdm 22339 | . . . . . 6 ⊢ (𝐷 ∈ (∞Met‘𝑥) → 𝑥 = dom dom 𝐷) | |
9 | 8 | fveq2d 6354 | . . . . 5 ⊢ (𝐷 ∈ (∞Met‘𝑥) → (∞Met‘𝑥) = (∞Met‘dom dom 𝐷)) |
10 | 7, 9 | eleqtrd 2839 | . . . 4 ⊢ (𝐷 ∈ (∞Met‘𝑥) → 𝐷 ∈ (∞Met‘dom dom 𝐷)) |
11 | 10 | rexlimivw 3165 | . . 3 ⊢ (∃𝑥 ∈ V 𝐷 ∈ (∞Met‘𝑥) → 𝐷 ∈ (∞Met‘dom dom 𝐷)) |
12 | 6, 11 | sylbi 207 | . 2 ⊢ (𝐷 ∈ ∪ ran ∞Met → 𝐷 ∈ (∞Met‘dom dom 𝐷)) |
13 | fvssunirn 6376 | . . 3 ⊢ (∞Met‘dom dom 𝐷) ⊆ ∪ ran ∞Met | |
14 | 13 | sseli 3738 | . 2 ⊢ (𝐷 ∈ (∞Met‘dom dom 𝐷) → 𝐷 ∈ ∪ ran ∞Met) |
15 | 12, 14 | impbii 199 | 1 ⊢ (𝐷 ∈ ∪ ran ∞Met ↔ 𝐷 ∈ (∞Met‘dom dom 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 ∧ wa 383 = wceq 1630 ∈ wcel 2137 ∀wral 3048 ∃wrex 3049 {crab 3052 Vcvv 3338 ∪ cuni 4586 class class class wbr 4802 × cxp 5262 dom cdm 5264 ran crn 5265 Fn wfn 6042 ‘cfv 6047 (class class class)co 6811 ↑𝑚 cmap 8021 0cc0 10126 ℝ*cxr 10263 ≤ cle 10265 +𝑒 cxad 12135 ∞Metcxmt 19931 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1869 ax-4 1884 ax-5 1986 ax-6 2052 ax-7 2088 ax-8 2139 ax-9 2146 ax-10 2166 ax-11 2181 ax-12 2194 ax-13 2389 ax-ext 2738 ax-sep 4931 ax-nul 4939 ax-pow 4990 ax-pr 5053 ax-un 7112 ax-cnex 10182 ax-resscn 10183 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1633 df-ex 1852 df-nf 1857 df-sb 2045 df-eu 2609 df-mo 2610 df-clab 2745 df-cleq 2751 df-clel 2754 df-nfc 2889 df-ne 2931 df-ral 3053 df-rex 3054 df-rab 3057 df-v 3340 df-sbc 3575 df-dif 3716 df-un 3718 df-in 3720 df-ss 3727 df-nul 4057 df-if 4229 df-pw 4302 df-sn 4320 df-pr 4322 df-op 4326 df-uni 4587 df-br 4803 df-opab 4863 df-mpt 4880 df-id 5172 df-xp 5270 df-rel 5271 df-cnv 5272 df-co 5273 df-dm 5274 df-rn 5275 df-iota 6010 df-fun 6049 df-fn 6050 df-f 6051 df-fv 6055 df-ov 6814 df-oprab 6815 df-mpt2 6816 df-map 8023 df-xr 10268 df-xmet 19939 |
This theorem is referenced by: isxms2 22452 setsmstopn 22482 tngtopn 22653 cfili 23264 cfilfcls 23270 |
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