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

Theorem ercl 6687
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 6685 . . . 4  |-  ( R  Er  X  ->  Rel  R )
31, 2syl 15 . . 3  |-  ( ph  ->  Rel  R )
4 ersym.2 . . 3  |-  ( ph  ->  A R B )
5 releldm 4927 . . 3  |-  ( ( Rel  R  /\  A R B )  ->  A  e.  dom  R )
63, 4, 5syl2anc 642 . 2  |-  ( ph  ->  A  e.  dom  R
)
7 erdm 6686 . . 3  |-  ( R  Er  X  ->  dom  R  =  X )
81, 7syl 15 . 2  |-  ( ph  ->  dom  R  =  X )
96, 8eleqtrd 2372 1  |-  ( ph  ->  A  e.  X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696   class class class wbr 4039   dom cdm 4705   Rel wrel 4710    Er wer 6673
This theorem is referenced by:  ercl2  6689  erthi  6722  qliftfun  6759  efgcpbl2  15082  frgpcpbl  15084  prter3  26853
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-xp 4711  df-rel 4712  df-dm 4715  df-er 6676
  Copyright terms: Public domain W3C validator