Theorem elcnv2 5786

Description: Membership in a converse relation. Equation 5 of [Suppes] p. 62. (Contributed by NM, 11-Aug-2004.)

Assertion

Ref	Expression
elcnv2	⊢ (𝐴 ∈ ◡𝑅 ↔ ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑦, 𝑥⟩ ∈ 𝑅))

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝑅,𝑦

Proof of Theorem elcnv2

Step	Hyp	Ref	Expression
1		elcnv 5785	. 2 ⊢ (𝐴 ∈ ◡𝑅 ↔ ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝑦𝑅𝑥))
2		df-br 5075	. . . 4 ⊢ (𝑦𝑅𝑥 ↔ ⟨𝑦, 𝑥⟩ ∈ 𝑅)
3	2	anbi2i 623	. . 3 ⊢ ((𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝑦𝑅𝑥) ↔ (𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑦, 𝑥⟩ ∈ 𝑅))
4	3	2exbii 1851	. 2 ⊢ (∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝑦𝑅𝑥) ↔ ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑦, 𝑥⟩ ∈ 𝑅))
5	1, 4	bitri 274	1 ⊢ (𝐴 ∈ ◡𝑅 ↔ ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑦, 𝑥⟩ ∈ 𝑅))

Syntax hints:   ↔ wb 205   ∧ wa 396   = wceq 1539  ∃wex 1782   ∈ wcel 2106  ⟨cop 4567   class class class wbr 5074  ^◡ccnv 5588

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-v 3434  df-dif 3890  df-un 3892  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-opab 5137  df-cnv 5597

This theorem is referenced by:  cnvuni  5795  elcnvlem  41209