Theorem cnvcnv3 6156

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 5642	. 2 ⊢ ◡◡𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝑦◡𝑅𝑥}
2		vex 3446	. . . 4 ⊢ 𝑦 ∈ V
3		vex 3446	. . . 4 ⊢ 𝑥 ∈ V
4	2, 3	brcnv 5841	. . 3 ⊢ (𝑦◡𝑅𝑥 ↔ 𝑥𝑅𝑦)
5	4	opabbii 5167	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝑦◡𝑅𝑥} = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦}
6	1, 5	eqtri 2760	1 ⊢ ◡◡𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦}

Syntax hints:   = wceq 1542   class class class wbr 5100  {copab 5162  ^◡ccnv 5633

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5245  ax-pr 5381

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-opab 5163  df-cnv 5642

This theorem is referenced by:  dfrel4v  6158