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Theorem brcnvg 4432
Description: The converse of a binary relation swaps arguments. Theorem 11 of [Suppes] p. 61. (Contributed by NM, 10-Oct-2005.)
Assertion
Ref Expression
brcnvg ((A 𝐶 B 𝐷) → (A𝑅BB𝑅A))

Proof of Theorem brcnvg
StepHypRef Expression
1 opelcnvg 4431 . 2 ((A 𝐶 B 𝐷) → (⟨A, B 𝑅 ↔ ⟨B, A 𝑅))
2 df-br 3729 . 2 (A𝑅B ↔ ⟨A, B 𝑅)
3 df-br 3729 . 2 (B𝑅A ↔ ⟨B, A 𝑅)
41, 2, 33bitr4g 212 1 ((A 𝐶 B 𝐷) → (A𝑅BB𝑅A))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   wcel 1367  cop 3343   class class class wbr 3728  ccnv 4260
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 614  ax-5 1310  ax-7 1311  ax-gen 1312  ax-ie1 1356  ax-ie2 1357  ax-8 1369  ax-10 1370  ax-11 1371  ax-i12 1372  ax-bnd 1373  ax-4 1374  ax-14 1379  ax-17 1393  ax-i9 1397  ax-ial 1401  ax-i5r 1402  ax-ext 1996  ax-sep 3839  ax-pow 3891  ax-pr 3908
This theorem depends on definitions:  df-bi 110  df-3an 869  df-tru 1227  df-nf 1324  df-sb 1620  df-eu 1877  df-mo 1878  df-clab 2001  df-cleq 2007  df-clel 2010  df-nfc 2141  df-v 2529  df-un 2891  df-in 2893  df-ss 2900  df-pw 3326  df-sn 3346  df-pr 3347  df-op 3349  df-br 3729  df-opab 3783  df-cnv 4269
This theorem is referenced by:  brcnv  4434  brelrng  4481  eliniseg  4611  relbrcnvg  4620  brcodir  4628  sefvex  5110  foeqcnvco  5344  isocnv2  5366  ersym  6018  brdifun  6033  ecidg  6070
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