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Theorem psmetrel 12566
 Description: The class of pseudometrics is a relation. (Contributed by Jim Kingdon, 24-Apr-2023.)
Assertion
Ref Expression
psmetrel Rel PsMet

Proof of Theorem psmetrel
Dummy variables 𝑤 𝑑 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mptrel 4679 . 2 Rel (𝑥 ∈ V ↦ {𝑑 ∈ (ℝ*𝑚 (𝑥 × 𝑥)) ∣ ∀𝑦𝑥 ((𝑦𝑑𝑦) = 0 ∧ ∀𝑧𝑥𝑤𝑥 (𝑦𝑑𝑧) ≤ ((𝑤𝑑𝑦) +𝑒 (𝑤𝑑𝑧)))})
2 df-psmet 12231 . . 3 PsMet = (𝑥 ∈ V ↦ {𝑑 ∈ (ℝ*𝑚 (𝑥 × 𝑥)) ∣ ∀𝑦𝑥 ((𝑦𝑑𝑦) = 0 ∧ ∀𝑧𝑥𝑤𝑥 (𝑦𝑑𝑧) ≤ ((𝑤𝑑𝑦) +𝑒 (𝑤𝑑𝑧)))})
32releqi 4634 . 2 (Rel PsMet ↔ Rel (𝑥 ∈ V ↦ {𝑑 ∈ (ℝ*𝑚 (𝑥 × 𝑥)) ∣ ∀𝑦𝑥 ((𝑦𝑑𝑦) = 0 ∧ ∀𝑧𝑥𝑤𝑥 (𝑦𝑑𝑧) ≤ ((𝑤𝑑𝑦) +𝑒 (𝑤𝑑𝑧)))}))
41, 3mpbir 145 1 Rel PsMet
 Colors of variables: wff set class Syntax hints:   ∧ wa 103   = wceq 1332  ∀wral 2418  {crab 2422  Vcvv 2691   class class class wbr 3939   ↦ cmpt 3999   × cxp 4549  Rel wrel 4556  (class class class)co 5786   ↑𝑚 cmap 6554  0cc0 7673  ℝ*cxr 7852   ≤ cle 7854   +𝑒 cxad 9616  PsMetcpsmet 12223 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2123  ax-sep 4056  ax-pow 4108  ax-pr 4142 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1738  df-clab 2128  df-cleq 2134  df-clel 2137  df-nfc 2272  df-ral 2423  df-rex 2424  df-v 2693  df-un 3082  df-in 3084  df-ss 3091  df-pw 3519  df-sn 3540  df-pr 3541  df-op 3543  df-opab 4000  df-mpt 4001  df-xp 4557  df-rel 4558  df-psmet 12231 This theorem is referenced by:  blfvalps  12629  blvalps  12632  blfps  12653
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