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Theorem ercl 6408
Description: Elementhood in the field of an equivalence relation. (Contributed by Mario Carneiro, 12-Aug-2015.)
Hypotheses
Ref Expression
ersym.1  |-  ( ph  ->  R  Er  X )
ersym.2  |-  ( ph  ->  A R B )
Assertion
Ref Expression
ercl  |-  ( ph  ->  A  e.  X )

Proof of Theorem ercl
StepHypRef Expression
1 ersym.1 . . . 4  |-  ( ph  ->  R  Er  X )
2 errel 6406 . . . 4  |-  ( R  Er  X  ->  Rel  R )
31, 2syl 14 . . 3  |-  ( ph  ->  Rel  R )
4 ersym.2 . . 3  |-  ( ph  ->  A R B )
5 releldm 4744 . . 3  |-  ( ( Rel  R  /\  A R B )  ->  A  e.  dom  R )
63, 4, 5syl2anc 408 . 2  |-  ( ph  ->  A  e.  dom  R
)
7 erdm 6407 . . 3  |-  ( R  Er  X  ->  dom  R  =  X )
81, 7syl 14 . 2  |-  ( ph  ->  dom  R  =  X )
96, 8eleqtrd 2196 1  |-  ( ph  ->  A  e.  X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1316    e. wcel 1465   class class class wbr 3899   dom cdm 4509   Rel wrel 4514    Er wer 6394
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-br 3900  df-opab 3960  df-xp 4515  df-rel 4516  df-dm 4519  df-er 6397
This theorem is referenced by:  ercl2  6410  erthi  6443  qliftfun  6479
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