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Theorem errn 6886
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 4848 . 2  |-  ran  R  =  dom  `' R
2 ercnv 6885 . . . 4  |-  ( R  Er  A  ->  `' R  =  R )
32dmeqd 5031 . . 3  |-  ( R  Er  A  ->  dom  `' R  =  dom  R
)
4 erdm 6874 . . 3  |-  ( R  Er  A  ->  dom  R  =  A )
53, 4eqtrd 2436 . 2  |-  ( R  Er  A  ->  dom  `' R  =  A )
61, 5syl5eq 2448 1  |-  ( R  Er  A  ->  ran  R  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649   `'ccnv 4836   dom cdm 4837   ran crn 4838    Er wer 6861
This theorem is referenced by:  erssxp  6887  ecss  6905  uniqs2  6925  sylow2a  15208
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-br 4173  df-opab 4227  df-xp 4843  df-rel 4844  df-cnv 4845  df-dm 4847  df-rn 4848  df-er 6864
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