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Theorem errn 6451
 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 4550 . 2 ran 𝑅 = dom 𝑅
2 ercnv 6450 . . . 4 (𝑅 Er 𝐴𝑅 = 𝑅)
32dmeqd 4741 . . 3 (𝑅 Er 𝐴 → dom 𝑅 = dom 𝑅)
4 erdm 6439 . . 3 (𝑅 Er 𝐴 → dom 𝑅 = 𝐴)
53, 4eqtrd 2172 . 2 (𝑅 Er 𝐴 → dom 𝑅 = 𝐴)
61, 5syl5eq 2184 1 (𝑅 Er 𝐴 → ran 𝑅 = 𝐴)
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1331  ◡ccnv 4538  dom cdm 4539  ran crn 4540   Er wer 6426 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131 This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-opab 3990  df-xp 4545  df-rel 4546  df-cnv 4547  df-dm 4549  df-rn 4550  df-er 6429 This theorem is referenced by:  erssxp  6452  ecss  6470  uniqs2  6489
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