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Theorem ereldm 6907
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 2580 . . 3  |-  ( ph  ->  ( [ A ] R  =/=  (/)  <->  [ B ] R  =/=  (/) ) )
3 ecdmn0 6906 . . 3  |-  ( A  e.  dom  R  <->  [ A ] R  =/=  (/) )
4 ecdmn0 6906 . . 3  |-  ( B  e.  dom  R  <->  [ B ] R  =/=  (/) )
52, 3, 43bitr4g 280 . 2  |-  ( ph  ->  ( A  e.  dom  R  <-> 
B  e.  dom  R
) )
6 ereldm.1 . . . 4  |-  ( ph  ->  R  Er  X )
7 erdm 6874 . . . 4  |-  ( R  Er  X  ->  dom  R  =  X )
86, 7syl 16 . . 3  |-  ( ph  ->  dom  R  =  X )
98eleq2d 2471 . 2  |-  ( ph  ->  ( A  e.  dom  R  <-> 
A  e.  X ) )
108eleq2d 2471 . 2  |-  ( ph  ->  ( B  e.  dom  R  <-> 
B  e.  X ) )
115, 9, 103bitr3d 275 1  |-  ( ph  ->  ( A  e.  X  <->  B  e.  X ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    = wceq 1649    e. wcel 1721    =/= wne 2567   (/)c0 3588   dom cdm 4837    Er wer 6861   [cec 6862
This theorem is referenced by:  erth  6908  brecop  6956  eceqoveq  6968
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-br 4173  df-opab 4227  df-xp 4843  df-cnv 4845  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-er 6864  df-ec 6866
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