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Mirrors > Home > ILE Home > Th. List > relcnv | GIF version |
Description: A converse is a relation. Theorem 12 of [Suppes] p. 62. (Contributed by NM, 29-Oct-1996.) |
Ref | Expression |
---|---|
relcnv | ⊢ Rel ◡𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-cnv 4446 | . 2 ⊢ ◡𝐴 = {〈𝑥, 𝑦〉 ∣ 𝑦𝐴𝑥} | |
2 | 1 | relopabi 4563 | 1 ⊢ Rel ◡𝐴 |
Colors of variables: wff set class |
Syntax hints: class class class wbr 3845 ◡ccnv 4437 Rel wrel 4443 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-14 1450 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 ax-sep 3957 ax-pow 4009 ax-pr 4036 |
This theorem depends on definitions: df-bi 115 df-3an 926 df-tru 1292 df-nf 1395 df-sb 1693 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-ral 2364 df-rex 2365 df-v 2621 df-un 3003 df-in 3005 df-ss 3012 df-pw 3431 df-sn 3452 df-pr 3453 df-op 3455 df-opab 3900 df-xp 4444 df-rel 4445 df-cnv 4446 |
This theorem is referenced by: relbrcnvg 4811 cnvsym 4815 intasym 4816 asymref 4817 cnvopab 4833 cnv0 4835 cnvdif 4838 dfrel2 4881 cnvcnv 4883 cnvsn0 4899 cnvcnvsn 4907 resdm2 4921 coi2 4947 coires1 4948 cnvssrndm 4952 unidmrn 4963 cnvexg 4968 cnviinm 4972 funi 5046 funcnvsn 5059 funcnv2 5074 funcnveq 5077 fcnvres 5194 f1cnvcnv 5227 f1ompt 5450 fliftcnv 5574 cnvf1o 5990 reldmtpos 6018 dmtpos 6021 rntpos 6022 dftpos3 6027 dftpos4 6028 tpostpos 6029 tposf12 6034 ercnv 6311 cnvct 6524 relcnvfi 6648 fsumcnv 10827 fisumcom2 10828 |
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