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Theorem psmetdmdm 22157
 Description: Recover the base set from a pseudometric. (Contributed by Thierry Arnoux, 7-Feb-2018.)
Assertion
Ref Expression
psmetdmdm (𝐷 ∈ (PsMet‘𝑋) → 𝑋 = dom dom 𝐷)

Proof of Theorem psmetdmdm
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elfvex 6259 . . . . 5 (𝐷 ∈ (PsMet‘𝑋) → 𝑋 ∈ V)
2 ispsmet 22156 . . . . . 6 (𝑋 ∈ V → (𝐷 ∈ (PsMet‘𝑋) ↔ (𝐷:(𝑋 × 𝑋)⟶ℝ* ∧ ∀𝑥𝑋 ((𝑥𝐷𝑥) = 0 ∧ ∀𝑦𝑋𝑧𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦))))))
32biimpa 500 . . . . 5 ((𝑋 ∈ V ∧ 𝐷 ∈ (PsMet‘𝑋)) → (𝐷:(𝑋 × 𝑋)⟶ℝ* ∧ ∀𝑥𝑋 ((𝑥𝐷𝑥) = 0 ∧ ∀𝑦𝑋𝑧𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦)))))
41, 3mpancom 704 . . . 4 (𝐷 ∈ (PsMet‘𝑋) → (𝐷:(𝑋 × 𝑋)⟶ℝ* ∧ ∀𝑥𝑋 ((𝑥𝐷𝑥) = 0 ∧ ∀𝑦𝑋𝑧𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦)))))
54simpld 474 . . 3 (𝐷 ∈ (PsMet‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*)
6 fdm 6089 . . . 4 (𝐷:(𝑋 × 𝑋)⟶ℝ* → dom 𝐷 = (𝑋 × 𝑋))
76dmeqd 5358 . . 3 (𝐷:(𝑋 × 𝑋)⟶ℝ* → dom dom 𝐷 = dom (𝑋 × 𝑋))
85, 7syl 17 . 2 (𝐷 ∈ (PsMet‘𝑋) → dom dom 𝐷 = dom (𝑋 × 𝑋))
9 dmxpid 5377 . 2 dom (𝑋 × 𝑋) = 𝑋
108, 9syl6req 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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