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Theorem errn 8767
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 5696 . 2 ran 𝑅 = dom 𝑅
2 ercnv 8766 . . . 4 (𝑅 Er 𝐴𝑅 = 𝑅)
32dmeqd 5916 . . 3 (𝑅 Er 𝐴 → dom 𝑅 = dom 𝑅)
4 erdm 8755 . . 3 (𝑅 Er 𝐴 → dom 𝑅 = 𝐴)
53, 4eqtrd 2777 . 2 (𝑅 Er 𝐴 → dom 𝑅 = 𝐴)
61, 5eqtrid 2789 1 (𝑅 Er 𝐴 → ran 𝑅 = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  ccnv 5684  dom cdm 5685  ran crn 5686   Er wer 8742
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-xp 5691  df-rel 5692  df-cnv 5693  df-dm 5695  df-rn 5696  df-er 8745
This theorem is referenced by:  erssxp  8768  ecss  8793  uniqs2  8819  sylow2a  19637
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