ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  relcnv GIF version

Theorem relcnv 4731
Description: A converse is a relation. Theorem 12 of [Suppes] p. 62. (Contributed by NM, 29-Oct-1996.)
Assertion
Ref Expression
relcnv Rel 𝐴

Proof of Theorem relcnv
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-cnv 4381 . 2 𝐴 = {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐴𝑥}
21relopabi 4491 1 Rel 𝐴
Colors of variables: wff set class
Syntax hints:   class class class wbr 3792  ccnv 4372  Rel wrel 4378
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-opab 3847  df-xp 4379  df-rel 4380  df-cnv 4381
This theorem is referenced by:  relbrcnvg  4732  cnvsym  4736  intasym  4737  asymref  4738  cnvopab  4754  cnv0  4755  cnvdif  4758  dfrel2  4799  cnvcnv  4801  cnvsn0  4817  cnvcnvsn  4825  resdm2  4839  coi2  4865  coires1  4866  cnvssrndm  4870  unidmrn  4878  cnvexg  4883  cnviinm  4887  funi  4960  funcnvsn  4973  funcnv2  4987  funcnveq  4990  fcnvres  5101  f1cnvcnv  5128  f1ompt  5348  fliftcnv  5463  cnvf1o  5874  reldmtpos  5899  dmtpos  5902  rntpos  5903  dftpos3  5908  dftpos4  5909  tpostpos  5910  tposf12  5915  ercnv  6158
  Copyright terms: Public domain W3C validator