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Theorem elcnv 5867
Description: Membership in a converse relation. Equation 5 of [Suppes] p. 62. (Contributed by NM, 24-Mar-1998.)
Assertion
Ref Expression
elcnv (𝐴𝑅 ↔ ∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝑦𝑅𝑥))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝑅,𝑦

Proof of Theorem elcnv
StepHypRef Expression
1 df-cnv 5675 . . 3 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝑦𝑅𝑥}
21eleq2i 2817 . 2 (𝐴𝑅𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝑦𝑅𝑥})
3 elopab 5518 . 2 (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝑦𝑅𝑥} ↔ ∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝑦𝑅𝑥))
42, 3bitri 275 1 (𝐴𝑅 ↔ ∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝑦𝑅𝑥))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 395   = wceq 1533  wex 1773  wcel 2098  cop 4627   class class class wbr 5139  {copab 5201  ccnv 5666
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-opab 5202  df-cnv 5675
This theorem is referenced by:  elcnv2  5868  gsummpt2co  32693
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