Theorem cnvun 5033

Description: The converse of a union is the union of converses. Theorem 16 of [Suppes] p. 62. (The proof was shortened by Andrew Salmon, 27-Aug-2011.) (Contributed by set.mm contributors, 25-Mar-1998.) (Revised by set.mm contributors, 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		unopab 4638	. . 3 |- ({<.x, y>. | yAx} u. {<.x, y>. | yBx}) = {<.x, y>. | (yAx \/ yBx)}
2		brun 4692	. . . 4 |- (y(A u. B)x <-> (yAx \/ yBx))
3	2	opabbii 4626	. . 3 |- {<.x, y>. | y(A u. B)x} = {<.x, y>. | (yAx \/ yBx)}
4	1, 3	eqtr4i 2376	. 2 |- ({<.x, y>. | yAx} u. {<.x, y>. | yBx}) = {<.x, y>. | y(A u. B)x}
5		df-cnv 4785	. . 3 |- `'A = {<.x, y>. | yAx}
6		df-cnv 4785	. . 3 |- `'B = {<.x, y>. | yBx}
7	5, 6	uneq12i 3416	. 2 |- (`'A u. `'B) = ({<.x, y>. | yAx} u. {<.x, y>. | yBx})
8		df-cnv 4785	. 2 |- `'(A u. B) = {<.x, y>. | y(A u. B)x}
9	4, 7, 8	3eqtr4ri 2384	1 |- `'(A u. B) = (`'A u. `'B)

Syntax hints:   \/ wo 357   = wceq 1642   u. cun 3207  {copab 4622   class class class wbr 4639  `'ccnv 4771

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-nin 3211  df-compl 3212  df-un 3214  df-opab 4623  df-br 4640  df-cnv 4785

This theorem is referenced by:  rnun  5036  f1oun  5304