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Mirrors > Home > ILE Home > Th. List > psmetrel | Unicode version |
Description: The class of pseudometrics is a relation. (Contributed by Jim Kingdon, 24-Apr-2023.) |
Ref | Expression |
---|---|
psmetrel | PsMet |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mptrel 4726 | . 2 | |
2 | df-psmet 12528 | . . 3 PsMet | |
3 | 2 | releqi 4681 | . 2 PsMet |
4 | 1, 3 | mpbir 145 | 1 PsMet |
Colors of variables: wff set class |
Syntax hints: wa 103 wceq 1342 wral 2442 crab 2446 cvv 2721 class class class wbr 3976 cmpt 4037 cxp 4596 wrel 4603 (class class class)co 5836 cmap 6605 cc0 7744 cxr 7923 cle 7925 cxad 9697 PsMetcpsmet 12520 |
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 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-v 2723 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-opab 4038 df-mpt 4039 df-xp 4604 df-rel 4605 df-psmet 12528 |
This theorem is referenced by: blfvalps 12926 blvalps 12929 blfps 12950 |
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