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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 4517 | . 2 ⊢ ◡𝐴 = {〈𝑥, 𝑦〉 ∣ 𝑦𝐴𝑥} | |
2 | 1 | relopabi 4635 | 1 ⊢ Rel ◡𝐴 |
Colors of variables: wff set class |
Syntax hints: class class class wbr 3899 ◡ccnv 4508 Rel wrel 4514 |
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 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-opab 3960 df-xp 4515 df-rel 4516 df-cnv 4517 |
This theorem is referenced by: relbrcnvg 4888 cnvsym 4892 intasym 4893 asymref 4894 cnvopab 4910 cnv0 4912 cnvdif 4915 dfrel2 4959 cnvcnv 4961 cnvsn0 4977 cnvcnvsn 4985 resdm2 4999 coi2 5025 coires1 5026 cnvssrndm 5030 unidmrn 5041 cnvexg 5046 cnviinm 5050 funi 5125 funcnvsn 5138 funcnv2 5153 funcnveq 5156 fcnvres 5276 f1cnvcnv 5309 f1ompt 5539 fliftcnv 5664 cnvf1o 6090 reldmtpos 6118 dmtpos 6121 rntpos 6122 dftpos3 6127 dftpos4 6128 tpostpos 6129 tposf12 6134 ercnv 6418 cnvct 6671 relcnvfi 6797 fsumcnv 11174 fisumcom2 11175 |
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