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Theorem psmetdmdm 22891
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 6679 . . . . 5 (𝐷 ∈ (PsMet‘𝑋) → 𝑋 ∈ V)
2 ispsmet 22890 . . . . . 6 (𝑋 ∈ V → (𝐷 ∈ (PsMet‘𝑋) ↔ (𝐷:(𝑋 × 𝑋)⟶ℝ* ∧ ∀𝑥𝑋 ((𝑥𝐷𝑥) = 0 ∧ ∀𝑦𝑋𝑧𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦))))))
32biimpa 479 . . . . 5 ((𝑋 ∈ V ∧ 𝐷 ∈ (PsMet‘𝑋)) → (𝐷:(𝑋 × 𝑋)⟶ℝ* ∧ ∀𝑥𝑋 ((𝑥𝐷𝑥) = 0 ∧ ∀𝑦𝑋𝑧𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦)))))
41, 3mpancom 686 . . . 4 (𝐷 ∈ (PsMet‘𝑋) → (𝐷:(𝑋 × 𝑋)⟶ℝ* ∧ ∀𝑥𝑋 ((𝑥𝐷𝑥) = 0 ∧ ∀𝑦𝑋𝑧𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦)))))
54simpld 497 . . 3 (𝐷 ∈ (PsMet‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*)
6 fdm 6498 . . . 4 (𝐷:(𝑋 × 𝑋)⟶ℝ* → dom 𝐷 = (𝑋 × 𝑋))
76dmeqd 5750 . . 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 5776 . 2 dom (𝑋 × 𝑋) = 𝑋
108, 9syl6req 2872 1 (𝐷 ∈ (PsMet‘𝑋) → 𝑋 = dom dom 𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398   = wceq 1537  wcel 2114  wral 3125  Vcvv 3473   class class class wbr 5042   × cxp 5529  dom cdm 5531  wf 6327  cfv 6331  (class class class)co 7133  0cc0 10515  *cxr 10652  cle 10654   +𝑒 cxad 12484  PsMetcpsmet 20505
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-sep 5179  ax-nul 5186  ax-pow 5242  ax-pr 5306  ax-un 7439  ax-cnex 10571  ax-resscn 10572
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-ral 3130  df-rex 3131  df-rab 3134  df-v 3475  df-sbc 3753  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4270  df-if 4444  df-pw 4517  df-sn 4544  df-pr 4546  df-op 4550  df-uni 4815  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5436  df-xp 5537  df-rel 5538  df-cnv 5539  df-co 5540  df-dm 5541  df-rn 5542  df-iota 6290  df-fun 6333  df-fn 6334  df-f 6335  df-fv 6339  df-ov 7136  df-oprab 7137  df-mpo 7138  df-map 8386  df-xr 10657  df-psmet 20513
This theorem is referenced by:  blfvalps  22969  metuval  23135  metidval  31141  pstmval  31146
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