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Theorem relelrnb 4745
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 4698 . . 3  |-  ( A  e.  ran  R  -> 
( A  e.  ran  R  <->  E. x  x R A ) )
21ibi 175 . 2  |-  ( A  e.  ran  R  ->  E. x  x R A )
3 relelrn 4743 . . . 4  |-  ( ( Rel  R  /\  x R A )  ->  A  e.  ran  R )
43ex 114 . . 3  |-  ( Rel 
R  ->  ( x R A  ->  A  e. 
ran  R ) )
54exlimdv 1773 . 2  |-  ( Rel 
R  ->  ( E. x  x R A  ->  A  e.  ran  R ) )
62, 5impbid2 142 1  |-  ( Rel 
R  ->  ( A  e.  ran  R  <->  E. x  x R A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104   E.wex 1451    e. wcel 1463   class class class wbr 3897   ran crn 4508   Rel wrel 4512
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-br 3898  df-opab 3958  df-xp 4513  df-rel 4514  df-cnv 4515  df-dm 4517  df-rn 4518
This theorem is referenced by: (None)
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