Theorem cnvun 5188

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 4777	. . 3 |- `' ( A u. B ) = { <. x , y >. | y ( A u. B ) x }
2		unopab 4205	. . . 4 |- ( { <. x , y >. | y A x } u. { <. x , y >. | y B x } ) = { <. x , y >. | ( y A x \/ y B x ) }
3		brun 4177	. . . . 5 |- ( y ( A u. B ) x <-> ( y A x \/ y B x ) )
4	3	opabbii 4193	. . . 4 |- { <. x , y >. | y ( A u. B ) x } = { <. x , y >. | ( y A x \/ y B x ) }
5	2, 4	eqtr4i 2262	. . 3 |- ( { <. x , y >. | y A x } u. { <. x , y >. | y B x } ) = { <. x , y >. | y ( A u. B ) x }
6	1, 5	eqtr4i 2262	. 2 |- `' ( A u. B ) = ( { <. x , y >. | y A x } u. { <. x , y >. | y B x } )
7		df-cnv 4777	. . 3 |- `' A = { <. x , y >. | y A x }
8		df-cnv 4777	. . 3 |- `' B = { <. x , y >. | y B x }
9	7, 8	uneq12i 3381	. 2 |- ( `' A u. `' B ) = ( { <. x , y >. | y A x } u. { <. x , y >. | y B x } )
10	6, 9	eqtr4i 2262	1 |- `' ( A u. B ) = ( `' A u. `' B )

Syntax hints:   \/ wo 720   = wceq 1402   u. cun 3218   class class class wbr 4125   {copab 4186   `'ccnv 4768

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-un 3224  df-br 4126  df-opab 4188  df-cnv 4777

This theorem is referenced by:  rnun  5191  f1oun  5654  sbthlemi8  7271  caseinj  7419  djuinj  7436  xnn0nnen  10852