ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  psmetrel GIF version

Theorem psmetrel 14062
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 4767 . 2 Rel (𝑥 ∈ V ↦ {𝑑 ∈ (ℝ*𝑚 (𝑥 × 𝑥)) ∣ ∀𝑦𝑥 ((𝑦𝑑𝑦) = 0 ∧ ∀𝑧𝑥𝑤𝑥 (𝑦𝑑𝑧) ≤ ((𝑤𝑑𝑦) +𝑒 (𝑤𝑑𝑧)))})
2 df-psmet 13673 . . 3 PsMet = (𝑥 ∈ V ↦ {𝑑 ∈ (ℝ*𝑚 (𝑥 × 𝑥)) ∣ ∀𝑦𝑥 ((𝑦𝑑𝑦) = 0 ∧ ∀𝑧𝑥𝑤𝑥 (𝑦𝑑𝑧) ≤ ((𝑤𝑑𝑦) +𝑒 (𝑤𝑑𝑧)))})
32releqi 4721 . 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 1363  wral 2465  {crab 2469  Vcvv 2749   class class class wbr 4015  cmpt 4076   × cxp 4636  Rel wrel 4643  (class class class)co 5888  𝑚 cmap 6661  0cc0 7824  *cxr 8004  cle 8006   +𝑒 cxad 9783  PsMetcpsmet 13665
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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-v 2751  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-opab 4077  df-mpt 4078  df-xp 4644  df-rel 4645  df-psmet 13673
This theorem is referenced by:  blfvalps  14125  blvalps  14128  blfps  14149
  Copyright terms: Public domain W3C validator