Theorem brcnvg 5878

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 5152	. 2 ⊢ (𝑥 = 𝐴 → (𝑦𝑅𝑥 ↔ 𝑦𝑅𝐴))
2		breq1 5151	. 2 ⊢ (𝑦 = 𝐵 → (𝑦𝑅𝐴 ↔ 𝐵𝑅𝐴))
3		df-cnv 5684	. 2 ⊢ ◡𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝑦𝑅𝑥}
4	1, 2, 3	brabg 5539	1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴◡𝑅𝐵 ↔ 𝐵𝑅𝐴))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397   ∈ wcel 2107   class class class wbr 5148  ^◡ccnv 5675

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 5299  ax-nul 5306  ax-pr 5427

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 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-cnv 5684

This theorem is referenced by:  opelcnvg  5879  brcnv  5881  brelrng  5939  elinisegg  6090  relbrcnvg  6102  brcodir  6118  predep  6329  dffv2  6984  ersym  8712  brdifun  8729  eqinf  9476  inflb  9481  infglb  9482  infglbb  9483  infltoreq  9494  infempty  9499  brcnvtrclfv  14947  oduleg  18240  posglbdg  18365  znleval  21102  slenlt  27245  brbtwn  28147  fcoinvbr  31824  cnvordtrestixx  32882  xrge0iifiso  32904  orvcgteel  33455  fv1stcnv  34737  fv2ndcnv  34738  wsuclem  34786  wsuclb  34789  colineardim1  35022  eldmcnv  37203  ineccnvmo  37215  alrmomorn  37216  brcnvin  37229  brxrn  37233  dfcoss3  37273  cosscnv  37275  brcoss3  37292  brcosscnv  37331  cosscnvssid3  37335  cosscnvssid4  37336  brnonrel  42326  ntrneifv2  42817  glbprlem  47552  gte-lte  47723  gt-lt  47724