MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  errn Structured version   Visualization version   GIF version

Theorem errn 8294
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 5530 . 2 ran 𝑅 = dom 𝑅
2 ercnv 8293 . . . 4 (𝑅 Er 𝐴𝑅 = 𝑅)
32dmeqd 5738 . . 3 (𝑅 Er 𝐴 → dom 𝑅 = dom 𝑅)
4 erdm 8282 . . 3 (𝑅 Er 𝐴 → dom 𝑅 = 𝐴)
53, 4eqtrd 2833 . 2 (𝑅 Er 𝐴 → dom 𝑅 = 𝐴)
61, 5syl5eq 2845 1 (𝑅 Er 𝐴 → ran 𝑅 = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  ccnv 5518  dom cdm 5519  ran crn 5520   Er wer 8269
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-br 5031  df-opab 5093  df-xp 5525  df-rel 5526  df-cnv 5527  df-dm 5529  df-rn 5530  df-er 8272
This theorem is referenced by:  erssxp  8295  ecss  8318  uniqs2  8342  sylow2a  18736
  Copyright terms: Public domain W3C validator