Theorem cnvcnv3 6186

Description: The set of all ordered pairs in a class is the same as the double converse. (Contributed by Mario Carneiro, 16-Aug-2015.)

Assertion

Ref	Expression
cnvcnv3	⊢ ◡◡𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦}

Distinct variable group:   𝑥,𝑦,𝑅

Proof of Theorem cnvcnv3

Step	Hyp	Ref	Expression
1		df-cnv 5669	. 2 ⊢ ◡◡𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝑦◡𝑅𝑥}
2		vex 3457	. . . 4 ⊢ 𝑦 ∈ V
3		vex 3457	. . . 4 ⊢ 𝑥 ∈ V
4	2, 3	brcnv 5868	. . 3 ⊢ (𝑦◡𝑅𝑥 ↔ 𝑥𝑅𝑦)
5	4	opabbii 5177	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝑦◡𝑅𝑥} = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦}
6	1, 5	eqtri 2784	1 ⊢ ◡◡𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦}

Syntax hints:   = wceq 1568   class class class wbr 5108  {copab 5172  ^◡ccnv 5660

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-ext 2733  ax-sep 5256  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-sb 2095  df-clab 2740  df-cleq 2753  df-clel 2836  df-rab 3415  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-br 5109  df-opab 5173  df-cnv 5669

This theorem is referenced by:  dfrel4v  6188