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Theorem errn 6064
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 4299 . 2 ran 𝑅 = dom 𝑅
2 ercnv 6063 . . . 4 (𝑅 Er A𝑅 = 𝑅)
32dmeqd 4480 . . 3 (𝑅 Er A → dom 𝑅 = dom 𝑅)
4 erdm 6052 . . 3 (𝑅 Er A → dom 𝑅 = A)
53, 4eqtrd 2069 . 2 (𝑅 Er A → dom 𝑅 = A)
61, 5syl5eq 2081 1 (𝑅 Er A → ran 𝑅 = A)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1242  ccnv 4287  dom cdm 4288  ran crn 4289   Er wer 6039
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-xp 4294  df-rel 4295  df-cnv 4296  df-dm 4298  df-rn 4299  df-er 6042
This theorem is referenced by:  erssxp  6065  ecss  6083  uniqs2  6102
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