Theorem brcnvg 5853

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 5106	. 2 ⊢ (𝑥 = 𝐴 → (𝑦𝑅𝑥 ↔ 𝑦𝑅𝐴))
2		breq1 5105	. 2 ⊢ (𝑦 = 𝐵 → (𝑦𝑅𝐴 ↔ 𝐵𝑅𝐴))
3		df-cnv 5657	. 2 ⊢ ◡𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝑦𝑅𝑥}
4	1, 2, 3	brabg 5512	1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴◡𝑅𝐵 ↔ 𝐵𝑅𝐴))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   ∈ wcel 2144   class class class wbr 5102  ^◡ccnv 5648

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-ext 2736  ax-sep 5248  ax-pr 5392

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-sb 2093  df-clab 2743  df-cleq 2756  df-clel 2839  df-rab 3417  df-v 3458  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5103  df-opab 5165  df-cnv 5657

This theorem is referenced by:  opelcnvg  5854  brcnv  5856  brelrng  5919  elinisegg  6084  relbrcnvg  6096  brcodir  6108  predep  6319  dffv2  6964  ersym  8693  brdifun  8711  eqinf  9433  inflb  9438  infglb  9439  infglbb  9440  infltoreq  9452  infempty  9457  brcnvtrclfv  15018  oduleg  18324  posglbdg  18447  znleval  21608  lenlts  27818  tgelrnpln  28985  brbtwn  29102  fcoinvbr  32807  cnvordtrestixx  34212  xrge0iifiso  34234  orvcgteel  34767  fv1stcnv  36132  fv2ndcnv  36133  wsuclem  36178  wsuclb  36181  colineardim1  36416  eldmcnv  38849  ineccnvmo  38861  alrmomorn  38862  brcnvin  38882  brxrn  38887  dfcoss3  39008  cosscnv  39010  brcoss3  39027  brcosscnv  39066  cosscnvssid3  39070  cosscnvssid4  39071  brnonrel  44170  ntrneifv2  44661  glbprlem  49591  gte-lte  50350  gt-lt  50351