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Theorem errn 6956
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 4918 . 2  |-  ran  R  =  dom  `' R
2 ercnv 6955 . . . 4  |-  ( R  Er  A  ->  `' R  =  R )
32dmeqd 5101 . . 3  |-  ( R  Er  A  ->  dom  `' R  =  dom  R
)
4 erdm 6944 . . 3  |-  ( R  Er  A  ->  dom  R  =  A )
53, 4eqtrd 2474 . 2  |-  ( R  Er  A  ->  dom  `' R  =  A )
61, 5syl5eq 2486 1  |-  ( R  Er  A  ->  ran  R  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653   `'ccnv 4906   dom cdm 4907   ran crn 4908    Er wer 6931
This theorem is referenced by:  erssxp  6957  ecss  6975  uniqs2  6995  sylow2a  15284
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-sep 4355  ax-nul 4363  ax-pr 4432
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-rex 2717  df-rab 2720  df-v 2964  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-sn 3844  df-pr 3845  df-op 3847  df-br 4238  df-opab 4292  df-xp 4913  df-rel 4914  df-cnv 4915  df-dm 4917  df-rn 4918  df-er 6934
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