Theorem cnvun 5088

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 4683	. . 3 |- `' ( A u. B ) = { <. x , y >. | y ( A u. B ) x }
2		unopab 4123	. . . 4 |- ( { <. x , y >. | y A x } u. { <. x , y >. | y B x } ) = { <. x , y >. | ( y A x \/ y B x ) }
3		brun 4095	. . . . 5 |- ( y ( A u. B ) x <-> ( y A x \/ y B x ) )
4	3	opabbii 4111	. . . 4 |- { <. x , y >. | y ( A u. B ) x } = { <. x , y >. | ( y A x \/ y B x ) }
5	2, 4	eqtr4i 2229	. . 3 |- ( { <. x , y >. | y A x } u. { <. x , y >. | y B x } ) = { <. x , y >. | y ( A u. B ) x }
6	1, 5	eqtr4i 2229	. 2 |- `' ( A u. B ) = ( { <. x , y >. | y A x } u. { <. x , y >. | y B x } )
7		df-cnv 4683	. . 3 |- `' A = { <. x , y >. | y A x }
8		df-cnv 4683	. . 3 |- `' B = { <. x , y >. | y B x }
9	7, 8	uneq12i 3325	. 2 |- ( `' A u. `' B ) = ( { <. x , y >. | y A x } u. { <. x , y >. | y B x } )
10	6, 9	eqtr4i 2229	1 |- `' ( A u. B ) = ( `' A u. `' B )

Syntax hints:   \/ wo 710   = wceq 1373   u. cun 3164   class class class wbr 4044   {copab 4104   `'ccnv 4674

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-un 3170  df-br 4045  df-opab 4106  df-cnv 4683

This theorem is referenced by:  rnun  5091  f1oun  5542  sbthlemi8  7066  caseinj  7191  djuinj  7208  xnn0nnen  10582