Theorem brcnvg 5733

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 5043	. 2 ⊢ (𝑥 = 𝐴 → (𝑦𝑅𝑥 ↔ 𝑦𝑅𝐴))
2		breq1 5042	. 2 ⊢ (𝑦 = 𝐵 → (𝑦𝑅𝐴 ↔ 𝐵𝑅𝐴))
3		df-cnv 5544	. 2 ⊢ ◡𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝑦𝑅𝑥}
4	1, 2, 3	brabg 5405	1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴◡𝑅𝐵 ↔ 𝐵𝑅𝐴))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∈ wcel 2112   class class class wbr 5039  ^◡ccnv 5535

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-ext 2708  ax-sep 5177  ax-nul 5184  ax-pr 5307

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2073  df-clab 2715  df-cleq 2728  df-clel 2809  df-rab 3060  df-v 3400  df-dif 3856  df-un 3858  df-nul 4224  df-if 4426  df-sn 4528  df-pr 4530  df-op 4534  df-br 5040  df-opab 5102  df-cnv 5544

This theorem is referenced by:  opelcnvg  5734  brcnv  5736  brelrng  5795  eliniseg  5943  relbrcnvg  5953  brcodir  5964  elpredg  6154  predep  6166  dffv2  6784  ersym  8381  brdifun  8398  eqinf  9078  inflb  9083  infglb  9084  infglbb  9085  infltoreq  9096  infempty  9101  brcnvtrclfv  14531  oduleg  17752  posglbdg  17875  znleval  20473  brbtwn  26944  fcoinvbr  30620  cnvordtrestixx  31531  xrge0iifiso  31553  orvcgteel  32100  fv1stcnv  33421  fv2ndcnv  33422  wsuclem  33499  wsuclb  33502  slenlt  33641  colineardim1  34049  eldmcnv  36166  ineccnvmo  36175  alrmomorn  36176  brxrn  36190  dfcoss3  36226  brcoss3  36242  brcosscnv  36276  cosscnvssid3  36280  cosscnvssid4  36281  brnonrel  40814  ntrneifv2  41308  glbprlem  45875  gte-lte  46040  gt-lt  46041