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Theorem cnvun 4672
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  `'  u.  `'  u.  `'

Proof of Theorem cnvun
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-cnv 4296 . . 3  `'  u.  { <. ,  >.  |  u.  }
2 unopab 3827 . . . 4  { <. ,  >.  |  }  u.  { <. , 
>.  |  }  { <. ,  >.  |  }
3 brun 3801 . . . . 5  u.
43opabbii 3815 . . . 4  { <. ,  >.  |  u.  }  { <. ,  >.  |  }
52, 4eqtr4i 2060 . . 3  { <. ,  >.  |  }  u.  { <. , 
>.  |  }  { <. ,  >.  |  u.  }
61, 5eqtr4i 2060 . 2  `'  u.  { <. ,  >.  |  }  u.  {
<. ,  >.  |  }
7 df-cnv 4296 . . 3  `'  { <. , 
>.  |  }
8 df-cnv 4296 . . 3  `'  { <. , 
>.  |  }
97, 8uneq12i 3089 . 2  `'  u.  `'  { <. ,  >.  |  }  u.  { <. , 
>.  |  }
106, 9eqtr4i 2060 1  `'  u.  `'  u.  `'
Colors of variables: wff set class
Syntax hints:   wo 628   wceq 1242    u. cun 2909   class class class wbr 3755   {copab 3808   `'ccnv 4287
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-br 3756  df-opab 3810  df-cnv 4296
This theorem is referenced by:  rnun  4675  f1oun  5089
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