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Theorem psmetrel 14219
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 4770 . 2 Rel (𝑥 ∈ V ↦ {𝑑 ∈ (ℝ*𝑚 (𝑥 × 𝑥)) ∣ ∀𝑦𝑥 ((𝑦𝑑𝑦) = 0 ∧ ∀𝑧𝑥𝑤𝑥 (𝑦𝑑𝑧) ≤ ((𝑤𝑑𝑦) +𝑒 (𝑤𝑑𝑧)))})
2 df-psmet 13817 . . 3 PsMet = (𝑥 ∈ V ↦ {𝑑 ∈ (ℝ*𝑚 (𝑥 × 𝑥)) ∣ ∀𝑦𝑥 ((𝑦𝑑𝑦) = 0 ∧ ∀𝑧𝑥𝑤𝑥 (𝑦𝑑𝑧) ≤ ((𝑤𝑑𝑦) +𝑒 (𝑤𝑑𝑧)))})
32releqi 4724 . 2 (Rel PsMet ↔ Rel (𝑥 ∈ V ↦ {𝑑 ∈ (ℝ*𝑚 (𝑥 × 𝑥)) ∣ ∀𝑦𝑥 ((𝑦𝑑𝑦) = 0 ∧ ∀𝑧𝑥𝑤𝑥 (𝑦𝑑𝑧) ≤ ((𝑤𝑑𝑦) +𝑒 (𝑤𝑑𝑧)))}))
41, 3mpbir 146 1 Rel PsMet
Colors of variables: wff set class
Syntax hints:  wa 104   = wceq 1364  wral 2468  {crab 2472  Vcvv 2752   class class class wbr 4018  cmpt 4079   × cxp 4639  Rel wrel 4646  (class class class)co 5891  𝑚 cmap 6666  0cc0 7829  *cxr 8009  cle 8011   +𝑒 cxad 9788  PsMetcpsmet 13809
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4189  ax-pr 4224
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-opab 4080  df-mpt 4081  df-xp 4647  df-rel 4648  df-psmet 13817
This theorem is referenced by:  blfvalps  14282  blvalps  14285  blfps  14306
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