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Theorem ereldm 6333
Description: Equality of equivalence classes implies equivalence of domain membership. (Contributed by NM, 28-Jan-1996.) (Revised by Mario Carneiro, 12-Aug-2015.)
Hypotheses
Ref Expression
ereldm.1  |-  ( ph  ->  R  Er  X )
ereldm.2  |-  ( ph  ->  [ A ] R  =  [ B ] R
)
Assertion
Ref Expression
ereldm  |-  ( ph  ->  ( A  e.  X  <->  B  e.  X ) )

Proof of Theorem ereldm
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 ereldm.2 . . . . 5  |-  ( ph  ->  [ A ] R  =  [ B ] R
)
21eleq2d 2157 . . . 4  |-  ( ph  ->  ( x  e.  [ A ] R  <->  x  e.  [ B ] R ) )
32exbidv 1753 . . 3  |-  ( ph  ->  ( E. x  x  e.  [ A ] R 
<->  E. x  x  e. 
[ B ] R
) )
4 ecdmn0m 6332 . . 3  |-  ( A  e.  dom  R  <->  E. x  x  e.  [ A ] R )
5 ecdmn0m 6332 . . 3  |-  ( B  e.  dom  R  <->  E. x  x  e.  [ B ] R )
63, 4, 53bitr4g 221 . 2  |-  ( ph  ->  ( A  e.  dom  R  <-> 
B  e.  dom  R
) )
7 ereldm.1 . . . 4  |-  ( ph  ->  R  Er  X )
8 erdm 6300 . . . 4  |-  ( R  Er  X  ->  dom  R  =  X )
97, 8syl 14 . . 3  |-  ( ph  ->  dom  R  =  X )
109eleq2d 2157 . 2  |-  ( ph  ->  ( A  e.  dom  R  <-> 
A  e.  X ) )
119eleq2d 2157 . 2  |-  ( ph  ->  ( B  e.  dom  R  <-> 
B  e.  X ) )
126, 10, 113bitr3d 216 1  |-  ( ph  ->  ( A  e.  X  <->  B  e.  X ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103    = wceq 1289   E.wex 1426    e. wcel 1438   dom cdm 4438    Er wer 6287   [cec 6288
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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2841  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-br 3846  df-opab 3900  df-xp 4444  df-cnv 4446  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-er 6290  df-ec 6292
This theorem is referenced by:  erth  6334  brecop  6380
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