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Theorem releldmb 5096
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 5057 . . 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 5094 . . . 4  |-  ( ( Rel  R  /\  A R x )  ->  A  e.  dom  R )
43ex 424 . . 3  |-  ( Rel 
R  ->  ( A R x  ->  A  e. 
dom  R ) )
54exlimdv 1646 . 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 1550    e. wcel 1725   class class class wbr 4204   dom cdm 4870   Rel wrel 4875
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-opab 4259  df-xp 4876  df-rel 4877  df-dm 4880
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