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Theorem cnvin 4674
Description: Distributive law for converse over intersection. Theorem 15 of [Suppes] p. 62. (Contributed by NM, 25-Mar-1998.) (Revised by Mario Carneiro, 26-Jun-2014.)
Assertion
Ref Expression
cnvin  `'  i^i  `'  i^i  `'

Proof of Theorem cnvin
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-cnv 4296 . . 3  `'  i^i  { <. ,  >.  |  i^i  }
2 inopab 4411 . . . 4  { <. ,  >.  |  }  i^i  { <. , 
>.  |  }  { <. ,  >.  |  }
3 brin 3802 . . . . 5  i^i
43opabbii 3815 . . . 4  { <. ,  >.  |  i^i  }  { <. ,  >.  |  }
52, 4eqtr4i 2060 . . 3  { <. ,  >.  |  }  i^i  { <. , 
>.  |  }  { <. ,  >.  |  i^i  }
61, 5eqtr4i 2060 . 2  `'  i^i  { <. ,  >.  |  }  i^i  {
<. ,  >.  |  }
7 df-cnv 4296 . . 3  `'  { <. , 
>.  |  }
8 df-cnv 4296 . . 3  `'  { <. , 
>.  |  }
97, 8ineq12i 3130 . 2  `'  i^i  `'  { <. ,  >.  |  }  i^i  { <. , 
>.  |  }
106, 9eqtr4i 2060 1  `'  i^i  `'  i^i  `'
Colors of variables: wff set class
Syntax hints:   wa 97   wceq 1242    i^i cin 2910   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-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-xp 4294  df-rel 4295  df-cnv 4296
This theorem is referenced by:  rnin  4676  dminxp  4708  imainrect  4709  cnvcnv  4716
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