Theorem cnvcnv3 4713

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 4296	. 2 ⊢ ◡◡𝑅 = {⟨x, y⟩ ∣ y◡𝑅x}
2		vex 2554	. . . 4 ⊢ y ∈ V
3		vex 2554	. . . 4 ⊢ x ∈ V
4	2, 3	brcnv 4461	. . 3 ⊢ (y◡𝑅x ↔ x𝑅y)
5	4	opabbii 3815	. 2 ⊢ {⟨x, y⟩ ∣ y◡𝑅x} = {⟨x, y⟩ ∣ x𝑅y}
6	1, 5	eqtri 2057	1 ⊢ ◡◡𝑅 = {⟨x, y⟩ ∣ x𝑅y}

Syntax hints:   = wceq 1242   class class class wbr 3755  {copab 3808  ^◡ccnv 4287

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-cnv 4296

This theorem is referenced by:  dfrel4v  4715