Theorem brcnvg 5818

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) → (𝐴◡𝑅𝐵 ↔ 𝐵𝑅𝐴))

Proof of Theorem brcnvg

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		breq2 5093	. 2 ⊢ (𝑥 = 𝐴 → (𝑦𝑅𝑥 ↔ 𝑦𝑅𝐴))
2		breq1 5092	. 2 ⊢ (𝑦 = 𝐵 → (𝑦𝑅𝐴 ↔ 𝐵𝑅𝐴))
3		df-cnv 5622	. 2 ⊢ ◡𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝑦𝑅𝑥}
4	1, 2, 3	brabg 5477	1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴◡𝑅𝐵 ↔ 𝐵𝑅𝐴))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∈ wcel 2111   class class class wbr 5089  ^◡ccnv 5613

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-br 5090  df-opab 5152  df-cnv 5622

This theorem is referenced by:  opelcnvg  5819  brcnv  5821  brelrng  5880  elinisegg  6041  relbrcnvg  6053  brcodir  6065  predep  6277  dffv2  6917  ersym  8634  brdifun  8652  eqinf  9369  inflb  9374  infglb  9375  infglbb  9376  infltoreq  9388  infempty  9393  brcnvtrclfv  14910  oduleg  18196  posglbdg  18319  znleval  21491  slenlt  27691  brbtwn  28877  fcoinvbr  32585  cnvordtrestixx  33926  xrge0iifiso  33948  orvcgteel  34481  fv1stcnv  35821  fv2ndcnv  35822  wsuclem  35867  wsuclb  35870  colineardim1  36105  eldmcnv  38376  ineccnvmo  38388  alrmomorn  38389  brcnvin  38401  brxrn  38406  dfcoss3  38515  cosscnv  38517  brcoss3  38534  brcosscnv  38573  cosscnvssid3  38577  cosscnvssid4  38578  brnonrel  43681  ntrneifv2  44172  glbprlem  49064  gte-lte  49824  gt-lt  49825