Theorem brcnvg 5832

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 5636	. 2 ⊢ ◡𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝑦𝑅𝑥}
4	1, 2, 3	brabg 5491	1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴◡𝑅𝐵 ↔ 𝐵𝑅𝐴))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∈ wcel 2114   class class class wbr 5086  ^◡ccnv 5627

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 5232  ax-pr 5374

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 5636

This theorem is referenced by:  opelcnvg  5833  brcnv  5835  brelrng  5894  elinisegg  6056  relbrcnvg  6068  brcodir  6080  predep  6292  dffv2  6933  ersym  8653  brdifun  8671  eqinf  9395  inflb  9400  infglb  9401  infglbb  9402  infltoreq  9414  infempty  9419  brcnvtrclfv  14962  oduleg  18253  posglbdg  18376  znleval  21531  lenlts  27713  brbtwn  28965  fcoinvbr  32672  cnvordtrestixx  34054  xrge0iifiso  34076  orvcgteel  34609  fv1stcnv  35956  fv2ndcnv  35957  wsuclem  36002  wsuclb  36005  colineardim1  36240  eldmcnv  38663  ineccnvmo  38675  alrmomorn  38676  brcnvin  38696  brxrn  38701  dfcoss3  38822  cosscnv  38824  brcoss3  38841  brcosscnv  38880  cosscnvssid3  38884  cosscnvssid4  38885  brnonrel  44013  ntrneifv2  44504  glbprlem  49431  gte-lte  50190  gt-lt  50191