Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  cnvun Structured version   Unicode version

Theorem cnvun 5280
 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.
StepHypRef Expression
1 df-cnv 4889 . . 3
2 unopab 4287 . . . 4
3 brun 4261 . . . . 5
43opabbii 4275 . . . 4
52, 4eqtr4i 2461 . . 3
61, 5eqtr4i 2461 . 2
7 df-cnv 4889 . . 3
8 df-cnv 4889 . . 3
97, 8uneq12i 3501 . 2
106, 9eqtr4i 2461 1
 Colors of variables: wff set class Syntax hints:   wo 359   wceq 1653   cun 3320   class class class wbr 4215  copab 4268  ccnv 4880 This theorem is referenced by:  rnun  5283  f1oun  5697  f1oprswap  5720  sbthlem8  7227  domss2  7269  1sdom  7314  fpwwe2lem13  8522  strlemor1  13561  xpsc  13787  gsumzaddlem  15531  mbfres2  19540  constr2spthlem1  21599  constr3pthlem2  21648  ex-cnv  21750  funsnfsup  26757 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-v 2960  df-un 3327  df-br 4216  df-opab 4270  df-cnv 4889
 Copyright terms: Public domain W3C validator