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Theorem errn 6551
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 4634 . 2  |-  ran  R  =  dom  `' R
2 ercnv 6550 . . . 4  |-  ( R  Er  A  ->  `' R  =  R )
32dmeqd 4825 . . 3  |-  ( R  Er  A  ->  dom  `' R  =  dom  R
)
4 erdm 6539 . . 3  |-  ( R  Er  A  ->  dom  R  =  A )
53, 4eqtrd 2210 . 2  |-  ( R  Er  A  ->  dom  `' R  =  A )
61, 5eqtrid 2222 1  |-  ( R  Er  A  ->  ran  R  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1353   `'ccnv 4622   dom cdm 4623   ran crn 4624    Er wer 6526
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-br 4001  df-opab 4062  df-xp 4629  df-rel 4630  df-cnv 4631  df-dm 4633  df-rn 4634  df-er 6529
This theorem is referenced by:  erssxp  6552  ecss  6570  uniqs2  6589
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