ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ercl Unicode version

Theorem ercl 6176
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 6174 . . . 4  |-  ( R  Er  X  ->  Rel  R )
31, 2syl 14 . . 3  |-  ( ph  ->  Rel  R )
4 ersym.2 . . 3  |-  ( ph  ->  A R B )
5 releldm 4591 . . 3  |-  ( ( Rel  R  /\  A R B )  ->  A  e.  dom  R )
63, 4, 5syl2anc 403 . 2  |-  ( ph  ->  A  e.  dom  R
)
7 erdm 6175 . . 3  |-  ( R  Er  X  ->  dom  R  =  X )
81, 7syl 14 . 2  |-  ( ph  ->  dom  R  =  X )
96, 8eleqtrd 2158 1  |-  ( ph  ->  A  e.  X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1285    e. wcel 1434   class class class wbr 3787   dom cdm 4365   Rel wrel 4370    Er wer 6162
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3898  ax-pow 3950  ax-pr 3966
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-br 3788  df-opab 3842  df-xp 4371  df-rel 4372  df-dm 4375  df-er 6165
This theorem is referenced by:  ercl2  6178  erthi  6211  qliftfun  6247
  Copyright terms: Public domain W3C validator