Theorem brcnvg 5880

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 5153	. 2 ⊢ (𝑥 = 𝐴 → (𝑦𝑅𝑥 ↔ 𝑦𝑅𝐴))
2		breq1 5152	. 2 ⊢ (𝑦 = 𝐵 → (𝑦𝑅𝐴 ↔ 𝐵𝑅𝐴))
3		df-cnv 5685	. 2 ⊢ ◡𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝑦𝑅𝑥}
4	1, 2, 3	brabg 5540	1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴◡𝑅𝐵 ↔ 𝐵𝑅𝐴))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397   ∈ wcel 2107   class class class wbr 5149  ^◡ccnv 5676

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-cnv 5685

This theorem is referenced by:  opelcnvg  5881  brcnv  5883  brelrng  5941  elinisegg  6093  relbrcnvg  6105  brcodir  6121  predep  6332  dffv2  6987  ersym  8715  brdifun  8732  eqinf  9479  inflb  9484  infglb  9485  infglbb  9486  infltoreq  9497  infempty  9502  brcnvtrclfv  14950  oduleg  18243  posglbdg  18368  znleval  21110  slenlt  27255  brbtwn  28157  fcoinvbr  31836  cnvordtrestixx  32893  xrge0iifiso  32915  orvcgteel  33466  fv1stcnv  34748  fv2ndcnv  34749  wsuclem  34797  wsuclb  34800  colineardim1  35033  eldmcnv  37214  ineccnvmo  37226  alrmomorn  37227  brcnvin  37240  brxrn  37244  dfcoss3  37284  cosscnv  37286  brcoss3  37303  brcosscnv  37342  cosscnvssid3  37346  cosscnvssid4  37347  brnonrel  42340  ntrneifv2  42831  glbprlem  47598  gte-lte  47769  gt-lt  47770