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Theorem errn 6585
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 4658 . 2 ran 𝑅 = dom 𝑅
2 ercnv 6584 . . . 4 (𝑅 Er 𝐴𝑅 = 𝑅)
32dmeqd 4850 . . 3 (𝑅 Er 𝐴 → dom 𝑅 = dom 𝑅)
4 erdm 6573 . . 3 (𝑅 Er 𝐴 → dom 𝑅 = 𝐴)
53, 4eqtrd 2222 . 2 (𝑅 Er 𝐴 → dom 𝑅 = 𝐴)
61, 5eqtrid 2234 1 (𝑅 Er 𝐴 → ran 𝑅 = 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1364  ccnv 4646  dom cdm 4647  ran crn 4648   Er wer 6560
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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4139  ax-pow 4195  ax-pr 4230
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-pw 3595  df-sn 3616  df-pr 3617  df-op 3619  df-br 4022  df-opab 4083  df-xp 4653  df-rel 4654  df-cnv 4655  df-dm 4657  df-rn 4658  df-er 6563
This theorem is referenced by:  erssxp  6586  ecss  6606  uniqs2  6625
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