![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 6678 | . . . . 5 ⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝑋 ∈ V) | |
2 | ispsmet 22911 | . . . . . 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 480 | . . . . 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 687 | . . . 4 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (𝐷:(𝑋 × 𝑋)⟶ℝ* ∧ ∀𝑥 ∈ 𝑋 ((𝑥𝐷𝑥) = 0 ∧ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦))))) |
5 | 4 | simpld 498 | . . 3 ⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*) |
6 | fdm 6495 | . . . 4 ⊢ (𝐷:(𝑋 × 𝑋)⟶ℝ* → dom 𝐷 = (𝑋 × 𝑋)) | |
7 | 6 | dmeqd 5738 | . . 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 5764 | . 2 ⊢ dom (𝑋 × 𝑋) = 𝑋 | |
10 | 8, 9 | eqtr2di 2850 | 1 ⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝑋 = dom dom 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∀wral 3106 Vcvv 3441 class class class wbr 5030 × cxp 5517 dom cdm 5519 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 0cc0 10526 ℝ*cxr 10663 ≤ cle 10665 +𝑒 cxad 12493 PsMetcpsmet 20075 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-map 8391 df-xr 10668 df-psmet 20083 |
This theorem is referenced by: blfvalps 22990 metuval 23156 metidval 31243 pstmval 31248 |
Copyright terms: Public domain | W3C validator |