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Theorem ercl 6539
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 6537 . . . 4  |-  ( R  Er  X  ->  Rel  R )
31, 2syl 14 . . 3  |-  ( ph  ->  Rel  R )
4 ersym.2 . . 3  |-  ( ph  ->  A R B )
5 releldm 4857 . . 3  |-  ( ( Rel  R  /\  A R B )  ->  A  e.  dom  R )
63, 4, 5syl2anc 411 . 2  |-  ( ph  ->  A  e.  dom  R
)
7 erdm 6538 . . 3  |-  ( R  Er  X  ->  dom  R  =  X )
81, 7syl 14 . 2  |-  ( ph  ->  dom  R  =  X )
96, 8eleqtrd 2256 1  |-  ( ph  ->  A  e.  X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1353    e. wcel 2148   class class class wbr 4000   dom cdm 4622   Rel wrel 4627    Er wer 6525
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4205
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-br 4001  df-opab 4062  df-xp 4628  df-rel 4629  df-dm 4632  df-er 6528
This theorem is referenced by:  ercl2  6541  erthi  6574  qliftfun  6610
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