Theorem metrel 15445

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.

Step	Hyp	Ref	Expression
1		mptrel 4908	. 2 ⊢ Rel (𝑒 ∈ V ↦ {𝑑 ∈ (ℝ ↑𝑚 (𝑒 × 𝑒)) ∣ ∀𝑥 ∈ 𝑒 ∀𝑦 ∈ 𝑒 (((𝑥𝑑𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝑒 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) + (𝑧𝑑𝑦)))})
2		df-met 14884	. . 3 ⊢ Met = (𝑒 ∈ V ↦ {𝑑 ∈ (ℝ ↑𝑚 (𝑒 × 𝑒)) ∣ ∀𝑥 ∈ 𝑒 ∀𝑦 ∈ 𝑒 (((𝑥𝑑𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝑒 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) + (𝑧𝑑𝑦)))})
3	2	releqi 4858	. 2 ⊢ (Rel Met ↔ Rel (𝑒 ∈ V ↦ {𝑑 ∈ (ℝ ↑𝑚 (𝑒 × 𝑒)) ∣ ∀𝑥 ∈ 𝑒 ∀𝑦 ∈ 𝑒 (((𝑥𝑑𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝑒 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) + (𝑧𝑑𝑦)))}))
4	1, 3	mpbir 146	1 ⊢ Rel Met

This proof depends on syntax axioms:   ∧ wa 104   ↔ wb 105   = wceq 1402  ∀wral 2528  {crab 2532  Vcvv 2821   class class class wbr 4130   ↦ cmpt 4192   × cxp 4772  Rel wrel 4779  (class class class)co 6085   ↑_𝑚 cmap 6922  ℝcr 8178  0cc0 8179   + caddc 8182   ≤ cle 8361  Metcmet 14876

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-opab 4193  df-mpt 4194  df-xp 4780  df-rel 4781  df-met 14884

This theorem is used by:  metflem  15452  ismet2  15457