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Theorem errn 8741
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 5684 . 2 ran 𝑅 = dom 𝑅
2 ercnv 8740 . . . 4 (𝑅 Er 𝐴𝑅 = 𝑅)
32dmeqd 5903 . . 3 (𝑅 Er 𝐴 → dom 𝑅 = dom 𝑅)
4 erdm 8729 . . 3 (𝑅 Er 𝐴 → dom 𝑅 = 𝐴)
53, 4eqtrd 2768 . 2 (𝑅 Er 𝐴 → dom 𝑅 = 𝐴)
61, 5eqtrid 2780 1 (𝑅 Er 𝐴 → ran 𝑅 = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1534  ccnv 5672  dom cdm 5673  ran crn 5674   Er wer 8716
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699  ax-sep 5294  ax-nul 5301  ax-pr 5424
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3472  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4320  df-if 4526  df-sn 4626  df-pr 4628  df-op 4632  df-br 5144  df-opab 5206  df-xp 5679  df-rel 5680  df-cnv 5681  df-dm 5683  df-rn 5684  df-er 8719
This theorem is referenced by:  erssxp  8742  ecss  8766  uniqs2  8792  sylow2a  19568
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