MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  releldmb Structured version   Unicode version

Theorem releldmb 5080
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 5041 . . 3  |-  ( A  e.  dom  R  -> 
( A  e.  dom  R  <->  E. x  A R x ) )
21ibi 244 . 2  |-  ( A  e.  dom  R  ->  E. x  A R x )
3 releldm 5078 . . . 4  |-  ( ( Rel  R  /\  A R x )  ->  A  e.  dom  R )
43ex 435 . . 3  |-  ( Rel 
R  ->  ( A R x  ->  A  e. 
dom  R ) )
54exlimdv 1768 . 2  |-  ( Rel 
R  ->  ( E. x  A R x  ->  A  e.  dom  R ) )
62, 5impbid2 207 1  |-  ( Rel 
R  ->  ( A  e.  dom  R  <->  E. x  A R x ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187   E.wex 1659    e. wcel 1867   class class class wbr 4417   dom cdm 4845   Rel wrel 4850
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4539  ax-nul 4547  ax-pr 4652
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-rab 2782  df-v 3080  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-sn 3994  df-pr 3996  df-op 4000  df-br 4418  df-opab 4476  df-xp 4851  df-rel 4852  df-dm 4855
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator