Theorem cnvun 5134

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 4727	. . 3 |- `' ( A u. B ) = { <. x , y >. | y ( A u. B ) x }
2		unopab 4163	. . . 4 |- ( { <. x , y >. | y A x } u. { <. x , y >. | y B x } ) = { <. x , y >. | ( y A x \/ y B x ) }
3		brun 4135	. . . . 5 |- ( y ( A u. B ) x <-> ( y A x \/ y B x ) )
4	3	opabbii 4151	. . . 4 |- { <. x , y >. | y ( A u. B ) x } = { <. x , y >. | ( y A x \/ y B x ) }
5	2, 4	eqtr4i 2253	. . 3 |- ( { <. x , y >. | y A x } u. { <. x , y >. | y B x } ) = { <. x , y >. | y ( A u. B ) x }
6	1, 5	eqtr4i 2253	. 2 |- `' ( A u. B ) = ( { <. x , y >. | y A x } u. { <. x , y >. | y B x } )
7		df-cnv 4727	. . 3 |- `' A = { <. x , y >. | y A x }
8		df-cnv 4727	. . 3 |- `' B = { <. x , y >. | y B x }
9	7, 8	uneq12i 3356	. 2 |- ( `' A u. `' B ) = ( { <. x , y >. | y A x } u. { <. x , y >. | y B x } )
10	6, 9	eqtr4i 2253	1 |- `' ( A u. B ) = ( `' A u. `' B )

Syntax hints:   \/ wo 713   = wceq 1395   u. cun 3195   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-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-un 3201  df-br 4084  df-opab 4146  df-cnv 4727

This theorem is referenced by:  rnun  5137  f1oun  5592  sbthlemi8  7131  caseinj  7256  djuinj  7273  xnn0nnen  10659