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Mirrors > Home > ILE Home > Th. List > xmetrel | GIF version |
Description: The class of extended metrics is a relation. (Contributed by Jim Kingdon, 20-Apr-2023.) |
Ref | Expression |
---|---|
xmetrel | ⊢ Rel ∞Met |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mptrel 4662 | . 2 ⊢ Rel (𝑒 ∈ V ↦ {𝑑 ∈ (ℝ* ↑𝑚 (𝑒 × 𝑒)) ∣ ∀𝑥 ∈ 𝑒 ∀𝑦 ∈ 𝑒 (((𝑥𝑑𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝑒 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦)))}) | |
2 | df-xmet 12146 | . . 3 ⊢ ∞Met = (𝑒 ∈ V ↦ {𝑑 ∈ (ℝ* ↑𝑚 (𝑒 × 𝑒)) ∣ ∀𝑥 ∈ 𝑒 ∀𝑦 ∈ 𝑒 (((𝑥𝑑𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝑒 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦)))}) | |
3 | 2 | releqi 4617 | . 2 ⊢ (Rel ∞Met ↔ Rel (𝑒 ∈ V ↦ {𝑑 ∈ (ℝ* ↑𝑚 (𝑒 × 𝑒)) ∣ ∀𝑥 ∈ 𝑒 ∀𝑦 ∈ 𝑒 (((𝑥𝑑𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝑒 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦)))})) |
4 | 1, 3 | mpbir 145 | 1 ⊢ Rel ∞Met |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ↔ wb 104 = wceq 1331 ∀wral 2414 {crab 2418 Vcvv 2681 class class class wbr 3924 ↦ cmpt 3984 × cxp 4532 Rel wrel 4539 (class class class)co 5767 ↑𝑚 cmap 6535 0cc0 7613 ℝ*cxr 7792 ≤ cle 7794 +𝑒 cxad 9550 ∞Metcxmet 12138 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-v 2683 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-opab 3985 df-mpt 3986 df-xp 4540 df-rel 4541 df-xmet 12146 |
This theorem is referenced by: ismet2 12512 xmeteq0 12517 xmettri2 12519 xmetpsmet 12527 xmetres2 12537 blex 12545 blval 12547 blf 12568 mopnval 12600 comet 12657 |
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