Theorem brcnvg 5822

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 5096	. 2 ⊢ (𝑥 = 𝐴 → (𝑦𝑅𝑥 ↔ 𝑦𝑅𝐴))
2		breq1 5095	. 2 ⊢ (𝑦 = 𝐵 → (𝑦𝑅𝐴 ↔ 𝐵𝑅𝐴))
3		df-cnv 5627	. 2 ⊢ ◡𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝑦𝑅𝑥}
4	1, 2, 3	brabg 5482	1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴◡𝑅𝐵 ↔ 𝐵𝑅𝐴))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∈ wcel 2109   class class class wbr 5092  ^◡ccnv 5618

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pr 5371

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-rab 3395  df-v 3438  df-dif 3906  df-un 3908  df-ss 3920  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-br 5093  df-opab 5155  df-cnv 5627

This theorem is referenced by:  opelcnvg  5823  brcnv  5825  brelrng  5883  elinisegg  6044  relbrcnvg  6056  brcodir  6068  predep  6278  dffv2  6918  ersym  8637  brdifun  8655  eqinf  9375  inflb  9380  infglb  9381  infglbb  9382  infltoreq  9394  infempty  9399  brcnvtrclfv  14910  oduleg  18196  posglbdg  18319  znleval  21461  slenlt  27662  brbtwn  28848  fcoinvbr  32554  cnvordtrestixx  33896  xrge0iifiso  33918  orvcgteel  34452  fv1stcnv  35770  fv2ndcnv  35771  wsuclem  35819  wsuclb  35822  colineardim1  36055  eldmcnv  38333  ineccnvmo  38345  alrmomorn  38346  brcnvin  38358  brxrn  38362  dfcoss3  38411  cosscnv  38413  brcoss3  38430  brcosscnv  38469  cosscnvssid3  38473  cosscnvssid4  38474  brnonrel  43582  ntrneifv2  44073  glbprlem  48969  gte-lte  49729  gt-lt  49730