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  14954  oduleg  18245  posglbdg  18368  znleval  21542  lenlts  27728  brbtwn  28980  fcoinvbr  32688  cnvordtrestixx  34071  xrge0iifiso  34093  orvcgteel  34626  fv1stcnv  35973  fv2ndcnv  35974  wsuclem  36019  wsuclb  36022  colineardim1  36257  eldmcnv  38670  ineccnvmo  38682  alrmomorn  38683  brcnvin  38703  brxrn  38708  dfcoss3  38829  cosscnv  38831  brcoss3  38848  brcosscnv  38887  cosscnvssid3  38891  cosscnvssid4  38892  brnonrel  44024  ntrneifv2  44515  glbprlem  49442  gte-lte  50201  gt-lt  50202