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Theorem elrnmpt 4788
Description: The range of a function in maps-to notation. (Contributed by Mario Carneiro, 20-Feb-2015.)
Hypothesis
Ref Expression
rnmpt.1  |-  F  =  ( x  e.  A  |->  B )
Assertion
Ref Expression
elrnmpt  |-  ( C  e.  V  ->  ( C  e.  ran  F  <->  E. x  e.  A  C  =  B ) )
Distinct variable group:    x, C
Allowed substitution hints:    A( x)    B( x)    F( x)    V( x)

Proof of Theorem elrnmpt
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 eqeq1 2146 . . 3  |-  ( y  =  C  ->  (
y  =  B  <->  C  =  B ) )
21rexbidv 2438 . 2  |-  ( y  =  C  ->  ( E. x  e.  A  y  =  B  <->  E. x  e.  A  C  =  B ) )
3 rnmpt.1 . . 3  |-  F  =  ( x  e.  A  |->  B )
43rnmpt 4787 . 2  |-  ran  F  =  { y  |  E. x  e.  A  y  =  B }
52, 4elab2g 2831 1  |-  ( C  e.  V  ->  ( C  e.  ran  F  <->  E. x  e.  A  C  =  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    = wceq 1331    e. wcel 1480   E.wrex 2417    |-> cmpt 3989   ran crn 4540
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-opab 3990  df-mpt 3991  df-cnv 4547  df-dm 4549  df-rn 4550
This theorem is referenced by:  elrnmpt1s  4789  elrnmptdv  4793  elrnmpt2d  4794  fifo  6868
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