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Theorem errn 6128
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 4356 . 2  |-  ran  R  =  dom  `' R
2 ercnv 6127 . . . 4  |-  ( R  Er  A  ->  `' R  =  R )
32dmeqd 4537 . . 3  |-  ( R  Er  A  ->  dom  `' R  =  dom  R
)
4 erdm 6116 . . 3  |-  ( R  Er  A  ->  dom  R  =  A )
53, 4eqtrd 2072 . 2  |-  ( R  Er  A  ->  dom  `' R  =  A )
61, 5syl5eq 2084 1  |-  ( R  Er  A  ->  ran  R  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1243   `'ccnv 4344   dom cdm 4345   ran crn 4346    Er wer 6103
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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-xp 4351  df-rel 4352  df-cnv 4353  df-dm 4355  df-rn 4356  df-er 6106
This theorem is referenced by:  erssxp  6129  ecss  6147  uniqs2  6166
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