![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
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 | β’ Rel βMet |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mptrel 4756 | . 2 β’ Rel (π β V β¦ {π β (β* βπ (π Γ π)) β£ βπ₯ β π βπ¦ β π (((π₯ππ¦) = 0 β π₯ = π¦) β§ βπ§ β π (π₯ππ¦) β€ ((π§ππ₯) +π (π§ππ¦)))}) | |
2 | df-xmet 13451 | . . 3 β’ βMet = (π β V β¦ {π β (β* βπ (π Γ π)) β£ βπ₯ β π βπ¦ β π (((π₯ππ¦) = 0 β π₯ = π¦) β§ βπ§ β π (π₯ππ¦) β€ ((π§ππ₯) +π (π§ππ¦)))}) | |
3 | 2 | releqi 4710 | . 2 β’ (Rel βMet β Rel (π β V β¦ {π β (β* βπ (π Γ π)) β£ βπ₯ β π βπ¦ β π (((π₯ππ¦) = 0 β π₯ = π¦) β§ βπ§ β π (π₯ππ¦) β€ ((π§ππ₯) +π (π§ππ¦)))})) |
4 | 1, 3 | mpbir 146 | 1 β’ Rel βMet |
Colors of variables: wff set class |
Syntax hints: β§ wa 104 β wb 105 = wceq 1353 βwral 2455 {crab 2459 Vcvv 2738 class class class wbr 4004 β¦ cmpt 4065 Γ cxp 4625 Rel wrel 4632 (class class class)co 5875 βπ cmap 6648 0cc0 7811 β*cxr 7991 β€ cle 7993 +π cxad 9770 βMetcxmet 13443 |
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 4122 ax-pow 4175 ax-pr 4210 |
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 2740 df-un 3134 df-in 3136 df-ss 3143 df-pw 3578 df-sn 3599 df-pr 3600 df-op 3602 df-opab 4066 df-mpt 4067 df-xp 4633 df-rel 4634 df-xmet 13451 |
This theorem is referenced by: ismet2 13857 xmeteq0 13862 xmettri2 13864 xmetpsmet 13872 xmetres2 13882 blex 13890 blval 13892 blf 13913 mopnval 13945 comet 14002 |
Copyright terms: Public domain | W3C validator |