Theorem cnvcnv3 5193

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 4739	. 2 |- `' `' R = { <. x , y >. | y `' R x }
2		vex 2806	. . . 4 |- y e. _V
3		vex 2806	. . . 4 |- x e. _V
4	2, 3	brcnv 4919	. . 3 |- ( y `' R x <-> x R y )
5	4	opabbii 4161	. 2 |- { <. x , y >. | y `' R x } = { <. x , y >. | x R y }
6	1, 5	eqtri 2252	1 |- `' `' R = { <. x , y >. | x R y }

Syntax hints:   = wceq 1398   class class class wbr 4093   {copab 4154   `'ccnv 4730

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-br 4094  df-opab 4156  df-cnv 4739

This theorem is referenced by:  dfrel4v  5195