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Theorem ercl 6917
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 6915 . . . 4  |-  ( R  Er  X  ->  Rel  R )
31, 2syl 16 . . 3  |-  ( ph  ->  Rel  R )
4 ersym.2 . . 3  |-  ( ph  ->  A R B )
5 releldm 5103 . . 3  |-  ( ( Rel  R  /\  A R B )  ->  A  e.  dom  R )
63, 4, 5syl2anc 644 . 2  |-  ( ph  ->  A  e.  dom  R
)
7 erdm 6916 . . 3  |-  ( R  Er  X  ->  dom  R  =  X )
81, 7syl 16 . 2  |-  ( ph  ->  dom  R  =  X )
96, 8eleqtrd 2513 1  |-  ( ph  ->  A  e.  X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653    e. wcel 1726   class class class wbr 4213   dom cdm 4879   Rel wrel 4884    Er wer 6903
This theorem is referenced by:  ercl2  6919  erthi  6952  qliftfun  6990  efgcpbl2  15390  frgpcpbl  15392  prter3  26732
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pr 4404
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-op 3824  df-br 4214  df-opab 4268  df-xp 4885  df-rel 4886  df-dm 4889  df-er 6906
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