ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  errn Unicode version

Theorem errn 6451
Description: The range and domain of an equivalence relation are equal. (Contributed by Rodolfo Medina, 11-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.)
Assertion
Ref Expression
errn  |-  ( R  Er  A  ->  ran  R  =  A )

Proof of Theorem errn
StepHypRef Expression
1 df-rn 4550 . 2  |-  ran  R  =  dom  `' R
2 ercnv 6450 . . . 4  |-  ( R  Er  A  ->  `' R  =  R )
32dmeqd 4741 . . 3  |-  ( R  Er  A  ->  dom  `' R  =  dom  R
)
4 erdm 6439 . . 3  |-  ( R  Er  A  ->  dom  R  =  A )
53, 4eqtrd 2172 . 2  |-  ( R  Er  A  ->  dom  `' R  =  A )
61, 5syl5eq 2184 1  |-  ( R  Er  A  ->  ran  R  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1331   `'ccnv 4538   dom cdm 4539   ran crn 4540    Er wer 6426
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-opab 3990  df-xp 4545  df-rel 4546  df-cnv 4547  df-dm 4549  df-rn 4550  df-er 6429
This theorem is referenced by:  erssxp  6452  ecss  6470  uniqs2  6489
  Copyright terms: Public domain W3C validator