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Theorem ereldm 6472
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 2209 . . . 4  |-  ( ph  ->  ( x  e.  [ A ] R  <->  x  e.  [ B ] R ) )
32exbidv 1797 . . 3  |-  ( ph  ->  ( E. x  x  e.  [ A ] R 
<->  E. x  x  e. 
[ B ] R
) )
4 ecdmn0m 6471 . . 3  |-  ( A  e.  dom  R  <->  E. x  x  e.  [ A ] R )
5 ecdmn0m 6471 . . 3  |-  ( B  e.  dom  R  <->  E. x  x  e.  [ B ] R )
63, 4, 53bitr4g 222 . 2  |-  ( ph  ->  ( A  e.  dom  R  <-> 
B  e.  dom  R
) )
7 ereldm.1 . . . 4  |-  ( ph  ->  R  Er  X )
8 erdm 6439 . . . 4  |-  ( R  Er  X  ->  dom  R  =  X )
97, 8syl 14 . . 3  |-  ( ph  ->  dom  R  =  X )
109eleq2d 2209 . 2  |-  ( ph  ->  ( A  e.  dom  R  <-> 
A  e.  X ) )
119eleq2d 2209 . 2  |-  ( ph  ->  ( B  e.  dom  R  <-> 
B  e.  X ) )
126, 10, 113bitr3d 217 1  |-  ( ph  ->  ( A  e.  X  <->  B  e.  X ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    = wceq 1331   E.wex 1468    e. wcel 1480   dom cdm 4539    Er wer 6426   [cec 6427
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-sbc 2910  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-opab 3990  df-xp 4545  df-cnv 4547  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-er 6429  df-ec 6431
This theorem is referenced by:  erth  6473  brecop  6519
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