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Theorem elcnv2 5891
Description: Membership in a converse relation. Equation 5 of [Suppes] p. 62. (Contributed by NM, 11-Aug-2004.)
Assertion
Ref Expression
elcnv2 (𝐴𝑅 ↔ ∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑦, 𝑥⟩ ∈ 𝑅))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝑅,𝑦

Proof of Theorem elcnv2
StepHypRef Expression
1 elcnv 5890 . 2 (𝐴𝑅 ↔ ∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝑦𝑅𝑥))
2 df-br 5149 . . . 4 (𝑦𝑅𝑥 ↔ ⟨𝑦, 𝑥⟩ ∈ 𝑅)
32anbi2i 623 . . 3 ((𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝑦𝑅𝑥) ↔ (𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑦, 𝑥⟩ ∈ 𝑅))
432exbii 1846 . 2 (∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝑦𝑅𝑥) ↔ ∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑦, 𝑥⟩ ∈ 𝑅))
51, 4bitri 275 1 (𝐴𝑅 ↔ ∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑦, 𝑥⟩ ∈ 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1537  wex 1776  wcel 2106  cop 4637   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-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:  cnvuni  5900  elcnvlem  43591
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