Theorem cnvcnv3 6191

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 5663	. 2 ⊢ ◡◡𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝑦◡𝑅𝑥}
2		vex 3458	. . . 4 ⊢ 𝑦 ∈ V
3		vex 3458	. . . 4 ⊢ 𝑥 ∈ V
4	2, 3	brcnv 5866	. . 3 ⊢ (𝑦◡𝑅𝑥 ↔ 𝑥𝑅𝑦)
5	4	opabbii 5175	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝑦◡𝑅𝑥} = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦}
6	1, 5	eqtri 2786	1 ⊢ ◡◡𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦}

This proof depends on syntax axioms:   = wceq 1570   class class class wbr 5106  {copab 5170  ^◡ccnv 5654

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5252  ax-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-br 5107  df-opab 5171  df-cnv 5663

This theorem is used by:  dfrel4v  6193  bj-dfrel2  38146  bj-cnvcnv2  38147