Theorem cnvun 5107

Description: The converse of a union is the union of converses. Theorem 16 of [Suppes] p. 62. (Contributed by NM, 25-Mar-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Assertion

Ref	Expression
cnvun	|- `' ( A u. B ) = ( `' A u. `' B )

Proof of Theorem cnvun

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-cnv 4701	. . 3 |- `' ( A u. B ) = { <. x , y >. | y ( A u. B ) x }
2		unopab 4139	. . . 4 |- ( { <. x , y >. | y A x } u. { <. x , y >. | y B x } ) = { <. x , y >. | ( y A x \/ y B x ) }
3		brun 4111	. . . . 5 |- ( y ( A u. B ) x <-> ( y A x \/ y B x ) )
4	3	opabbii 4127	. . . 4 |- { <. x , y >. | y ( A u. B ) x } = { <. x , y >. | ( y A x \/ y B x ) }
5	2, 4	eqtr4i 2231	. . 3 |- ( { <. x , y >. | y A x } u. { <. x , y >. | y B x } ) = { <. x , y >. | y ( A u. B ) x }
6	1, 5	eqtr4i 2231	. 2 |- `' ( A u. B ) = ( { <. x , y >. | y A x } u. { <. x , y >. | y B x } )
7		df-cnv 4701	. . 3 |- `' A = { <. x , y >. | y A x }
8		df-cnv 4701	. . 3 |- `' B = { <. x , y >. | y B x }
9	7, 8	uneq12i 3333	. 2 |- ( `' A u. `' B ) = ( { <. x , y >. | y A x } u. { <. x , y >. | y B x } )
10	6, 9	eqtr4i 2231	1 |- `' ( A u. B ) = ( `' A u. `' B )

Syntax hints:   \/ wo 710   = wceq 1373   u. cun 3172   class class class wbr 4059   {copab 4120   `'ccnv 4692

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2778  df-un 3178  df-br 4060  df-opab 4122  df-cnv 4701

This theorem is referenced by:  rnun  5110  f1oun  5564  sbthlemi8  7092  caseinj  7217  djuinj  7234  xnn0nnen  10619