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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 4755 | . 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 13339 | . . 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 4709 | . 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 146 | 1 ⊢ Rel ∞Met |
Colors of variables: wff set class |
Syntax hints: ∧ wa 104 ↔ wb 105 = wceq 1353 ∀wral 2455 {crab 2459 Vcvv 2737 class class class wbr 4003 ↦ cmpt 4064 × cxp 4624 Rel wrel 4631 (class class class)co 5874 ↑𝑚 cmap 6647 0cc0 7810 ℝ*cxr 7989 ≤ cle 7991 +𝑒 cxad 9768 ∞Metcxmet 13331 |
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 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4121 ax-pow 4174 ax-pr 4209 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2739 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-opab 4065 df-mpt 4066 df-xp 4632 df-rel 4633 df-xmet 13339 |
This theorem is referenced by: ismet2 13747 xmeteq0 13752 xmettri2 13754 xmetpsmet 13762 xmetres2 13772 blex 13780 blval 13782 blf 13803 mopnval 13835 comet 13892 |
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