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Theorem relelrnb 5038
Description: Membership in a range. (Contributed by Mario Carneiro, 5-Nov-2015.)
Assertion
Ref Expression
relelrnb  |-  ( Rel 
R  ->  ( A  e.  ran  R  <->  E. x  x R A ) )
Distinct variable groups:    x, A    x, R

Proof of Theorem relelrnb
StepHypRef Expression
1 elrng 4995 . . 3  |-  ( A  e.  ran  R  -> 
( A  e.  ran  R  <->  E. x  x R A ) )
21ibi 233 . 2  |-  ( A  e.  ran  R  ->  E. x  x R A )
3 relelrn 5036 . . . 4  |-  ( ( Rel  R  /\  x R A )  ->  A  e.  ran  R )
43ex 424 . . 3  |-  ( Rel 
R  ->  ( x R A  ->  A  e. 
ran  R ) )
54exlimdv 1643 . 2  |-  ( Rel 
R  ->  ( E. x  x R A  ->  A  e.  ran  R ) )
62, 5impbid2 196 1  |-  ( Rel 
R  ->  ( A  e.  ran  R  <->  E. x  x R A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177   E.wex 1547    e. wcel 1717   class class class wbr 4146   ran crn 4812   Rel wrel 4816
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 2361  ax-sep 4264  ax-nul 4272  ax-pr 4337
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-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-op 3759  df-br 4147  df-opab 4201  df-xp 4817  df-rel 4818  df-cnv 4819  df-dm 4821  df-rn 4822
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