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Theorem errn 6312
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 4449 . 2  |-  ran  R  =  dom  `' R
2 ercnv 6311 . . . 4  |-  ( R  Er  A  ->  `' R  =  R )
32dmeqd 4638 . . 3  |-  ( R  Er  A  ->  dom  `' R  =  dom  R
)
4 erdm 6300 . . 3  |-  ( R  Er  A  ->  dom  R  =  A )
53, 4eqtrd 2120 . 2  |-  ( R  Er  A  ->  dom  `' R  =  A )
61, 5syl5eq 2132 1  |-  ( R  Er  A  ->  ran  R  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1289   `'ccnv 4437   dom cdm 4438   ran crn 4439    Er wer 6287
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-br 3846  df-opab 3900  df-xp 4444  df-rel 4445  df-cnv 4446  df-dm 4448  df-rn 4449  df-er 6290
This theorem is referenced by:  erssxp  6313  ecss  6331  uniqs2  6350
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