Theorem brcnvg 5893

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 5152	. 2 ⊢ (𝑥 = 𝐴 → (𝑦𝑅𝑥 ↔ 𝑦𝑅𝐴))
2		breq1 5151	. 2 ⊢ (𝑦 = 𝐵 → (𝑦𝑅𝐴 ↔ 𝐵𝑅𝐴))
3		df-cnv 5697	. 2 ⊢ ◡𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝑦𝑅𝑥}
4	1, 2, 3	brabg 5549	1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴◡𝑅𝐵 ↔ 𝐵𝑅𝐴))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∈ wcel 2106   class class class wbr 5148  ^◡ccnv 5688

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-cnv 5697

This theorem is referenced by:  opelcnvg  5894  brcnv  5896  brelrng  5955  elinisegg  6114  relbrcnvg  6126  brcodir  6142  predep  6353  dffv2  7004  ersym  8756  brdifun  8774  eqinf  9522  inflb  9527  infglb  9528  infglbb  9529  infltoreq  9540  infempty  9545  brcnvtrclfv  15039  oduleg  18347  posglbdg  18473  znleval  21591  slenlt  27812  brbtwn  28929  fcoinvbr  32625  cnvordtrestixx  33874  xrge0iifiso  33896  orvcgteel  34449  fv1stcnv  35758  fv2ndcnv  35759  wsuclem  35807  wsuclb  35810  colineardim1  36043  eldmcnv  38327  ineccnvmo  38339  alrmomorn  38340  brcnvin  38352  brxrn  38356  dfcoss3  38396  cosscnv  38398  brcoss3  38415  brcosscnv  38454  cosscnvssid3  38458  cosscnvssid4  38459  brnonrel  43579  ntrneifv2  44070  glbprlem  48762  gte-lte  48955  gt-lt  48956