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Theorem elcnv 5784
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 5598 . . 3 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝑦𝑅𝑥}
21eleq2i 2832 . 2 (𝐴𝑅𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝑦𝑅𝑥})
3 elopab 5443 . 2 (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝑦𝑅𝑥} ↔ ∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝑦𝑅𝑥))
42, 3bitri 274 1 (𝐴𝑅 ↔ ∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝑦𝑅𝑥))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 396   = wceq 1542  wex 1786  wcel 2110  cop 4573   class class class wbr 5079  {copab 5141  ccnv 5589
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-ext 2711  ax-sep 5227  ax-nul 5234  ax-pr 5356
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-sb 2072  df-clab 2718  df-cleq 2732  df-clel 2818  df-v 3433  df-dif 3895  df-un 3897  df-nul 4263  df-if 4466  df-sn 4568  df-pr 4570  df-op 4574  df-opab 5142  df-cnv 5598
This theorem is referenced by:  elcnv2  5785  gsummpt2co  31317
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