Theorem brcnvg 5890

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 5147	. 2 ⊢ (𝑥 = 𝐴 → (𝑦𝑅𝑥 ↔ 𝑦𝑅𝐴))
2		breq1 5146	. 2 ⊢ (𝑦 = 𝐵 → (𝑦𝑅𝐴 ↔ 𝐵𝑅𝐴))
3		df-cnv 5693	. 2 ⊢ ◡𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝑦𝑅𝑥}
4	1, 2, 3	brabg 5544	1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴◡𝑅𝐵 ↔ 𝐵𝑅𝐴))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∈ wcel 2108   class class class wbr 5143  ^◡ccnv 5684

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 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-cnv 5693

This theorem is referenced by:  opelcnvg  5891  brcnv  5893  brelrng  5952  elinisegg  6111  relbrcnvg  6123  brcodir  6139  predep  6351  dffv2  7004  ersym  8757  brdifun  8775  eqinf  9524  inflb  9529  infglb  9530  infglbb  9531  infltoreq  9542  infempty  9547  brcnvtrclfv  15042  oduleg  18335  posglbdg  18460  znleval  21573  slenlt  27797  brbtwn  28914  fcoinvbr  32618  cnvordtrestixx  33912  xrge0iifiso  33934  orvcgteel  34470  fv1stcnv  35777  fv2ndcnv  35778  wsuclem  35826  wsuclb  35829  colineardim1  36062  eldmcnv  38346  ineccnvmo  38358  alrmomorn  38359  brcnvin  38371  brxrn  38375  dfcoss3  38415  cosscnv  38417  brcoss3  38434  brcosscnv  38473  cosscnvssid3  38477  cosscnvssid4  38478  brnonrel  43602  ntrneifv2  44093  glbprlem  48862  gte-lte  49243  gt-lt  49244