Theorem metrel 14662

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 4795	. 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 14177	. . 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 4747	. 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

Syntax hints:   ∧ wa 104   ↔ wb 105   = wceq 1364  ∀wral 2475  {crab 2479  Vcvv 2763   class class class wbr 4034   ↦ cmpt 4095   × cxp 4662  Rel wrel 4669  (class class class)co 5925   ↑_𝑚 cmap 6716  ℝcr 7895  0cc0 7896   + caddc 7899   ≤ cle 8079  Metcmet 14169

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-opab 4096  df-mpt 4097  df-xp 4670  df-rel 4671  df-met 14177

This theorem is referenced by:  metflem  14669  ismet2  14674