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Theorem elcnv2 5880
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 5879 . 2 (𝐴𝑅 ↔ ∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝑦𝑅𝑥))
2 df-br 5149 . . . 4 (𝑦𝑅𝑥 ↔ ⟨𝑦, 𝑥⟩ ∈ 𝑅)
32anbi2i 622 . . 3 ((𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝑦𝑅𝑥) ↔ (𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑦, 𝑥⟩ ∈ 𝑅))
432exbii 1844 . 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 205  wa 395   = wceq 1534  wex 1774  wcel 2099  cop 4635   class class class wbr 5148  ccnv 5677
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5149  df-opab 5211  df-cnv 5686
This theorem is referenced by:  cnvuni  5889  elcnvlem  43031
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