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Mirrors > Home > MPE Home > Th. List > psmetdmdm | Structured version Visualization version GIF version |
Description: Recover the base set from a pseudometric. (Contributed by Thierry Arnoux, 7-Feb-2018.) |
Ref | Expression |
---|---|
psmetdmdm | ⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝑋 = dom dom 𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfvex 6259 | . . . . 5 ⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝑋 ∈ V) | |
2 | ispsmet 22156 | . . . . . 6 ⊢ (𝑋 ∈ V → (𝐷 ∈ (PsMet‘𝑋) ↔ (𝐷:(𝑋 × 𝑋)⟶ℝ* ∧ ∀𝑥 ∈ 𝑋 ((𝑥𝐷𝑥) = 0 ∧ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦)))))) | |
3 | 2 | biimpa 500 | . . . . 5 ⊢ ((𝑋 ∈ V ∧ 𝐷 ∈ (PsMet‘𝑋)) → (𝐷:(𝑋 × 𝑋)⟶ℝ* ∧ ∀𝑥 ∈ 𝑋 ((𝑥𝐷𝑥) = 0 ∧ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦))))) |
4 | 1, 3 | mpancom 704 | . . . 4 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (𝐷:(𝑋 × 𝑋)⟶ℝ* ∧ ∀𝑥 ∈ 𝑋 ((𝑥𝐷𝑥) = 0 ∧ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦))))) |
5 | 4 | simpld 474 | . . 3 ⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*) |
6 | fdm 6089 | . . . 4 ⊢ (𝐷:(𝑋 × 𝑋)⟶ℝ* → dom 𝐷 = (𝑋 × 𝑋)) | |
7 | 6 | dmeqd 5358 | . . 3 ⊢ (𝐷:(𝑋 × 𝑋)⟶ℝ* → dom dom 𝐷 = dom (𝑋 × 𝑋)) |
8 | 5, 7 | syl 17 | . 2 ⊢ (𝐷 ∈ (PsMet‘𝑋) → dom dom 𝐷 = dom (𝑋 × 𝑋)) |
9 | dmxpid 5377 | . 2 ⊢ dom (𝑋 × 𝑋) = 𝑋 | |
10 | 8, 9 | syl6req 2702 | 1 ⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝑋 = dom dom 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1523 ∈ wcel 2030 ∀wral 2941 Vcvv 3231 class class class wbr 4685 × cxp 5141 dom cdm 5143 ⟶wf 5922 ‘cfv 5926 (class class class)co 6690 0cc0 9974 ℝ*cxr 10111 ≤ cle 10113 +𝑒 cxad 11982 PsMetcpsmet 19778 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-cnex 10030 ax-resscn 10031 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-ral 2946 df-rex 2947 df-rab 2950 df-v 3233 df-sbc 3469 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-op 4217 df-uni 4469 df-br 4686 df-opab 4746 df-mpt 4763 df-id 5053 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-fv 5934 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-map 7901 df-xr 10116 df-psmet 19786 |
This theorem is referenced by: blfvalps 22235 metuval 22401 metidval 30061 pstmval 30066 |
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