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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 4649 | . 2 ⊢ ◡𝐴 = {〈𝑥, 𝑦〉 ∣ 𝑦𝐴𝑥} | |
2 | 1 | relopabi 4767 | 1 ⊢ Rel ◡𝐴 |
Colors of variables: wff set class |
Syntax hints: class class class wbr 4018 ◡ccnv 4640 Rel wrel 4646 |
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 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4189 ax-pr 4224 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-v 2754 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-opab 4080 df-xp 4647 df-rel 4648 df-cnv 4649 |
This theorem is referenced by: relbrcnvg 5022 eliniseg2 5023 cnvsym 5027 intasym 5028 asymref 5029 cnvopab 5045 cnv0 5047 cnvdif 5050 dfrel2 5094 cnvcnv 5096 cnvsn0 5112 cnvcnvsn 5120 resdm2 5134 coi2 5160 coires1 5161 cnvssrndm 5165 unidmrn 5176 cnvexg 5181 cnviinm 5185 funi 5264 funcnvsn 5277 funcnv2 5292 funcnveq 5295 fcnvres 5415 f1cnvcnv 5448 f1ompt 5684 fliftcnv 5813 cnvf1o 6245 reldmtpos 6273 dmtpos 6276 rntpos 6277 dftpos3 6282 dftpos4 6283 tpostpos 6284 tposf12 6289 ercnv 6575 cnvct 6830 relcnvfi 6965 fsumcnv 11472 fisumcom2 11473 fprodcnv 11660 fprodcom2fi 11661 |
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