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Theorem releldmb 5045
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 5006 . . 3  |-  ( A  e.  dom  R  -> 
( A  e.  dom  R  <->  E. x  A R x ) )
21ibi 233 . 2  |-  ( A  e.  dom  R  ->  E. x  A R x )
3 releldm 5043 . . . 4  |-  ( ( Rel  R  /\  A R x )  ->  A  e.  dom  R )
43ex 424 . . 3  |-  ( Rel 
R  ->  ( A R x  ->  A  e. 
dom  R ) )
54exlimdv 1643 . 2  |-  ( Rel 
R  ->  ( E. x  A R x  ->  A  e.  dom  R ) )
62, 5impbid2 196 1  |-  ( Rel 
R  ->  ( A  e.  dom  R  <->  E. x  A R x ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177   E.wex 1547    e. wcel 1717   class class class wbr 4154   dom cdm 4819   Rel wrel 4824
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 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pr 4345
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-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-op 3767  df-br 4155  df-opab 4209  df-xp 4825  df-rel 4826  df-dm 4829
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