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Theorem ereldm 6703
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
StepHypRef Expression
1 ereldm.2 . . . 4  |-  ( ph  ->  [ A ] R  =  [ B ] R
)
21neeq1d 2459 . . 3  |-  ( ph  ->  ( [ A ] R  =/=  (/)  <->  [ B ] R  =/=  (/) ) )
3 ecdmn0 6702 . . 3  |-  ( A  e.  dom  R  <->  [ A ] R  =/=  (/) )
4 ecdmn0 6702 . . 3  |-  ( B  e.  dom  R  <->  [ B ] R  =/=  (/) )
52, 3, 43bitr4g 279 . 2  |-  ( ph  ->  ( A  e.  dom  R  <-> 
B  e.  dom  R
) )
6 ereldm.1 . . . 4  |-  ( ph  ->  R  Er  X )
7 erdm 6670 . . . 4  |-  ( R  Er  X  ->  dom  R  =  X )
86, 7syl 15 . . 3  |-  ( ph  ->  dom  R  =  X )
98eleq2d 2350 . 2  |-  ( ph  ->  ( A  e.  dom  R  <-> 
A  e.  X ) )
108eleq2d 2350 . 2  |-  ( ph  ->  ( B  e.  dom  R  <-> 
B  e.  X ) )
115, 9, 103bitr3d 274 1  |-  ( ph  ->  ( A  e.  X  <->  B  e.  X ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1623    e. wcel 1684    =/= wne 2446   (/)c0 3455   dom cdm 4689    Er wer 6657   [cec 6658
This theorem is referenced by:  erth  6704  brecop  6751  eceqoveq  6763
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-xp 4695  df-cnv 4697  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-er 6660  df-ec 6662
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