Theorem cnvun 5416

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

Proof of Theorem cnvun

Dummy variables are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-cnv 5012	. . 3
2		unopab 4527	. . . 4
3		brun 4500	. . . . 5
4	3	opabbii 4516	. . . 4
5	2, 4	eqtr4i 2489	. . 3
6	1, 5	eqtr4i 2489	. 2
7		df-cnv 5012	. . 3
8		df-cnv 5012	. . 3
9	7, 8	uneq12i 3655	. 2
10	6, 9	eqtr4i 2489	1

Syntax hints:  \/wo 368  =wceq 1395  u.cun 3473   class class class wbr 4452  {copab 4509  `'ccnv 5003

This theorem is referenced by:  rnun  5419  f1oun  5840  f1oprswap  5860  suppun  6939  sbthlem8  7654  domss2  7696  1sdom  7742  fsuppun  7868  fpwwe2lem13  9041  strlemor1  14724  xpsc  14954  gsumzaddlemOLD  16936  funsnfsupOLD  18256  mbfres2  22052  constr2spthlem1  24596  constr3pthlem2  24656  ex-cnv  25158  eulerpartlemt  28310  mthmpps  28942  trclubg  37785

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-v 3111  df-un 3480  df-br 4453  df-opab 4511  df-cnv 5012