Theorem brcnvg 5826

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 5090	. 2 ⊢ (𝑥 = 𝐴 → (𝑦𝑅𝑥 ↔ 𝑦𝑅𝐴))
2		breq1 5089	. 2 ⊢ (𝑦 = 𝐵 → (𝑦𝑅𝐴 ↔ 𝐵𝑅𝐴))
3		df-cnv 5630	. 2 ⊢ ◡𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝑦𝑅𝑥}
4	1, 2, 3	brabg 5485	1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴◡𝑅𝐵 ↔ 𝐵𝑅𝐴))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∈ wcel 2114   class class class wbr 5086  ^◡ccnv 5621

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5231  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-br 5087  df-opab 5149  df-cnv 5630

This theorem is referenced by:  opelcnvg  5827  brcnv  5829  brelrng  5888  elinisegg  6050  relbrcnvg  6062  brcodir  6074  predep  6286  dffv2  6927  ersym  8647  brdifun  8665  eqinf  9389  inflb  9394  infglb  9395  infglbb  9396  infltoreq  9408  infempty  9413  brcnvtrclfv  14927  oduleg  18214  posglbdg  18337  znleval  21511  lenlts  27704  brbtwn  28956  fcoinvbr  32664  cnvordtrestixx  34063  xrge0iifiso  34085  orvcgteel  34618  fv1stcnv  35965  fv2ndcnv  35966  wsuclem  36011  wsuclb  36014  colineardim1  36249  eldmcnv  38657  ineccnvmo  38669  alrmomorn  38670  brcnvin  38690  brxrn  38695  dfcoss3  38816  cosscnv  38818  brcoss3  38835  brcosscnv  38874  cosscnvssid3  38878  cosscnvssid4  38879  brnonrel  44019  ntrneifv2  44510  glbprlem  49398  gte-lte  50157  gt-lt  50158