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Theorem metrel 12586
 Description: The class of metrics is a relation. (Contributed by Jim Kingdon, 20-Apr-2023.)
Assertion
Ref Expression
metrel Rel Met

Proof of Theorem metrel
Dummy variables 𝑒 𝑑 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mptrel 4679 . 2 Rel (𝑒 ∈ V ↦ {𝑑 ∈ (ℝ ↑𝑚 (𝑒 × 𝑒)) ∣ ∀𝑥𝑒𝑦𝑒 (((𝑥𝑑𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧𝑒 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) + (𝑧𝑑𝑦)))})
2 df-met 12233 . . 3 Met = (𝑒 ∈ V ↦ {𝑑 ∈ (ℝ ↑𝑚 (𝑒 × 𝑒)) ∣ ∀𝑥𝑒𝑦𝑒 (((𝑥𝑑𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧𝑒 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) + (𝑧𝑑𝑦)))})
32releqi 4634 . 2 (Rel Met ↔ Rel (𝑒 ∈ V ↦ {𝑑 ∈ (ℝ ↑𝑚 (𝑒 × 𝑒)) ∣ ∀𝑥𝑒𝑦𝑒 (((𝑥𝑑𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧𝑒 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) + (𝑧𝑑𝑦)))}))
41, 3mpbir 145 1 Rel Met
 Colors of variables: wff set class Syntax hints:   ∧ wa 103   ↔ wb 104   = 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  ℝcr 7672  0cc0 7673   + caddc 7676   ≤ cle 7854  Metcmet 12225 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 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-14 2115  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 1732  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-met 12233 This theorem is referenced by:  metflem  12593  ismet2  12598
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