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

Theorem errn 6769
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 4782 . 2  |-  ran  R  =  dom  `' R
2 ercnv 6768 . . . 4  |-  ( R  Er  A  ->  `' R  =  R )
32dmeqd 4963 . . 3  |-  ( R  Er  A  ->  dom  `' R  =  dom  R
)
4 erdm 6757 . . 3  |-  ( R  Er  A  ->  dom  R  =  A )
53, 4eqtrd 2390 . 2  |-  ( R  Er  A  ->  dom  `' R  =  A )
61, 5syl5eq 2402 1  |-  ( R  Er  A  ->  ran  R  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1642   `'ccnv 4770   dom cdm 4771   ran crn 4772    Er wer 6744
This theorem is referenced by:  erssxp  6770  ecss  6788  uniqs2  6808  sylow2a  15029
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pr 4295
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-br 4105  df-opab 4159  df-xp 4777  df-rel 4778  df-cnv 4779  df-dm 4781  df-rn 4782  df-er 6747
  Copyright terms: Public domain W3C validator