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Theorem errn 8164
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 5457 . 2 ran 𝑅 = dom 𝑅
2 ercnv 8163 . . . 4 (𝑅 Er 𝐴𝑅 = 𝑅)
32dmeqd 5663 . . 3 (𝑅 Er 𝐴 → dom 𝑅 = dom 𝑅)
4 erdm 8152 . . 3 (𝑅 Er 𝐴 → dom 𝑅 = 𝐴)
53, 4eqtrd 2830 . 2 (𝑅 Er 𝐴 → dom 𝑅 = 𝐴)
61, 5syl5eq 2842 1 (𝑅 Er 𝐴 → ran 𝑅 = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1522  ccnv 5445  dom cdm 5446  ran crn 5447   Er wer 8139
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1778  ax-4 1792  ax-5 1889  ax-6 1948  ax-7 1993  ax-8 2082  ax-9 2090  ax-10 2111  ax-11 2125  ax-12 2140  ax-13 2343  ax-ext 2768  ax-sep 5097  ax-nul 5104  ax-pr 5224
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1763  df-nf 1767  df-sb 2042  df-mo 2575  df-eu 2611  df-clab 2775  df-cleq 2787  df-clel 2862  df-nfc 2934  df-ral 3109  df-rex 3110  df-rab 3113  df-v 3438  df-dif 3864  df-un 3866  df-in 3868  df-ss 3876  df-nul 4214  df-if 4384  df-sn 4475  df-pr 4477  df-op 4481  df-br 4965  df-opab 5027  df-xp 5452  df-rel 5453  df-cnv 5454  df-dm 5456  df-rn 5457  df-er 8142
This theorem is referenced by:  erssxp  8165  ecss  8188  uniqs2  8212  sylow2a  18474
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