Theorem cnvcnv3 5178

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 >. | x R y }

Distinct variable group:   x, y, R

Proof of Theorem cnvcnv3

Step	Hyp	Ref	Expression
1		df-cnv 4727	. 2 |- `' `' R = { <. x , y >. | y `' R x }
2		vex 2802	. . . 4 |- y e. _V
3		vex 2802	. . . 4 |- x e. _V
4	2, 3	brcnv 4905	. . 3 |- ( y `' R x <-> x R y )
5	4	opabbii 4151	. 2 |- { <. x , y >. | y `' R x } = { <. x , y >. | x R y }
6	1, 5	eqtri 2250	1 |- `' `' R = { <. x , y >. | x R y }

Syntax hints:   = wceq 1395   class class class wbr 4083   {copab 4144   `'ccnv 4718

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-br 4084  df-opab 4146  df-cnv 4727

This theorem is referenced by:  dfrel4v  5180