Theorem cnvun 5076

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 4672	. . 3 |- `' ( A u. B ) = { <. x , y >. | y ( A u. B ) x }
2		unopab 4113	. . . 4 |- ( { <. x , y >. | y A x } u. { <. x , y >. | y B x } ) = { <. x , y >. | ( y A x \/ y B x ) }
3		brun 4085	. . . . 5 |- ( y ( A u. B ) x <-> ( y A x \/ y B x ) )
4	3	opabbii 4101	. . . 4 |- { <. x , y >. | y ( A u. B ) x } = { <. x , y >. | ( y A x \/ y B x ) }
5	2, 4	eqtr4i 2220	. . 3 |- ( { <. x , y >. | y A x } u. { <. x , y >. | y B x } ) = { <. x , y >. | y ( A u. B ) x }
6	1, 5	eqtr4i 2220	. 2 |- `' ( A u. B ) = ( { <. x , y >. | y A x } u. { <. x , y >. | y B x } )
7		df-cnv 4672	. . 3 |- `' A = { <. x , y >. | y A x }
8		df-cnv 4672	. . 3 |- `' B = { <. x , y >. | y B x }
9	7, 8	uneq12i 3316	. 2 |- ( `' A u. `' B ) = ( { <. x , y >. | y A x } u. { <. x , y >. | y B x } )
10	6, 9	eqtr4i 2220	1 |- `' ( A u. B ) = ( `' A u. `' B )

Syntax hints:   \/ wo 709   = wceq 1364   u. cun 3155   class class class wbr 4034   {copab 4094   `'ccnv 4663

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-un 3161  df-br 4035  df-opab 4096  df-cnv 4672

This theorem is referenced by:  rnun  5079  f1oun  5527  sbthlemi8  7039  caseinj  7164  djuinj  7181  xnn0nnen  10546