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Theorem releldmb 4929
Description: Membership in a domain. (Contributed by Mario Carneiro, 5-Nov-2015.)
Assertion
Ref Expression
releldmb  |-  ( Rel 
R  ->  ( A  e.  dom  R  <->  E. x  A R x ) )
Distinct variable groups:    x, A    x, R

Proof of Theorem releldmb
StepHypRef Expression
1 eldmg 4890 . . 3  |-  ( A  e.  dom  R  -> 
( A  e.  dom  R  <->  E. x  A R x ) )
21ibi 232 . 2  |-  ( A  e.  dom  R  ->  E. x  A R x )
3 releldm 4927 . . . 4  |-  ( ( Rel  R  /\  A R x )  ->  A  e.  dom  R )
43ex 423 . . 3  |-  ( Rel 
R  ->  ( A R x  ->  A  e. 
dom  R ) )
54exlimdv 1626 . 2  |-  ( Rel 
R  ->  ( E. x  A R x  ->  A  e.  dom  R ) )
62, 5impbid2 195 1  |-  ( Rel 
R  ->  ( A  e.  dom  R  <->  E. x  A R x ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176   E.wex 1531    e. wcel 1696   class class class wbr 4039   dom cdm 4705   Rel wrel 4710
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-xp 4711  df-rel 4712  df-dm 4715
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