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Theorem relelrnb 4600
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 4554 . . 3  |-  ( A  e.  ran  R  -> 
( A  e.  ran  R  <->  E. x  x R A ) )
21ibi 174 . 2  |-  ( A  e.  ran  R  ->  E. x  x R A )
3 relelrn 4598 . . . 4  |-  ( ( Rel  R  /\  x R A )  ->  A  e.  ran  R )
43ex 113 . . 3  |-  ( Rel 
R  ->  ( x R A  ->  A  e. 
ran  R ) )
54exlimdv 1741 . 2  |-  ( Rel 
R  ->  ( E. x  x R A  ->  A  e.  ran  R ) )
62, 5impbid2 141 1  |-  ( Rel 
R  ->  ( A  e.  ran  R  <->  E. x  x R A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103   E.wex 1422    e. wcel 1434   class class class wbr 3793   ran crn 4372   Rel wrel 4376
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-br 3794  df-opab 3848  df-xp 4377  df-rel 4378  df-cnv 4379  df-dm 4381  df-rn 4382
This theorem is referenced by: (None)
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