Theorem cnvcnv3 5582

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 5122	. 2 ⊢ ◡◡𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝑦◡𝑅𝑥}
2		vex 3203	. . . 4 ⊢ 𝑦 ∈ V
3		vex 3203	. . . 4 ⊢ 𝑥 ∈ V
4	2, 3	brcnv 5305	. . 3 ⊢ (𝑦◡𝑅𝑥 ↔ 𝑥𝑅𝑦)
5	4	opabbii 4717	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝑦◡𝑅𝑥} = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦}
6	1, 5	eqtri 2644	1 ⊢ ◡◡𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦}

Syntax hints:   = wceq 1483   class class class wbr 4653  {copab 4712  ^◡ccnv 5113

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-cnv 5122

This theorem is referenced by:  dfrel4v  5584