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Theorem releldmb 5237
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 5198 . . 3  |-  ( A  e.  dom  R  -> 
( A  e.  dom  R  <->  E. x  A R x ) )
21ibi 241 . 2  |-  ( A  e.  dom  R  ->  E. x  A R x )
3 releldm 5235 . . . 4  |-  ( ( Rel  R  /\  A R x )  ->  A  e.  dom  R )
43ex 434 . . 3  |-  ( Rel 
R  ->  ( A R x  ->  A  e. 
dom  R ) )
54exlimdv 1700 . 2  |-  ( Rel 
R  ->  ( E. x  A R x  ->  A  e.  dom  R ) )
62, 5impbid2 204 1  |-  ( Rel 
R  ->  ( A  e.  dom  R  <->  E. x  A R x ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184   E.wex 1596    e. wcel 1767   class class class wbr 4447   dom cdm 4999   Rel wrel 5004
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-opab 4506  df-xp 5005  df-rel 5006  df-dm 5009
This theorem is referenced by: (None)
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