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Theorem relelrnb 4847
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 4800 . . 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 4845 . . . 4  |-  ( ( Rel  R  /\  x R A )  ->  A  e.  ran  R )
43ex 114 . . 3  |-  ( Rel 
R  ->  ( x R A  ->  A  e. 
ran  R ) )
54exlimdv 1812 . 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 1485    e. wcel 2141   class class class wbr 3987   ran crn 4610   Rel wrel 4614
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-br 3988  df-opab 4049  df-xp 4615  df-rel 4616  df-cnv 4617  df-dm 4619  df-rn 4620
This theorem is referenced by: (None)
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