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Theorem relelrnb 4864
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 4817 . . 3  |-  ( A  e.  ran  R  -> 
( A  e.  ran  R  <->  E. x  x R A ) )
21ibi 176 . 2  |-  ( A  e.  ran  R  ->  E. x  x R A )
3 relelrn 4862 . . . 4  |-  ( ( Rel  R  /\  x R A )  ->  A  e.  ran  R )
43ex 115 . . 3  |-  ( Rel 
R  ->  ( x R A  ->  A  e. 
ran  R ) )
54exlimdv 1819 . 2  |-  ( Rel 
R  ->  ( E. x  x R A  ->  A  e.  ran  R ) )
62, 5impbid2 143 1  |-  ( Rel 
R  ->  ( A  e.  ran  R  <->  E. x  x R A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 105   E.wex 1492    e. wcel 2148   class class class wbr 4002   ran crn 4626   Rel wrel 4630
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4120  ax-pow 4173  ax-pr 4208
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-br 4003  df-opab 4064  df-xp 4631  df-rel 4632  df-cnv 4633  df-dm 4635  df-rn 4636
This theorem is referenced by: (None)
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