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Theorem elrnmpti 4615
Description: Membership in the range of a function. (Contributed by NM, 30-Aug-2004.) (Revised by Mario Carneiro, 31-Aug-2015.)
Hypotheses
Ref Expression
rnmpt.1  |-  F  =  ( x  e.  A  |->  B )
elrnmpti.2  |-  B  e. 
_V
Assertion
Ref Expression
elrnmpti  |-  ( C  e.  ran  F  <->  E. x  e.  A  C  =  B )
Distinct variable group:    x, C
Allowed substitution hints:    A( x)    B( x)    F( x)

Proof of Theorem elrnmpti
StepHypRef Expression
1 elrnmpti.2 . . 3  |-  B  e. 
_V
21rgenw 2419 . 2  |-  A. x  e.  A  B  e.  _V
3 rnmpt.1 . . 3  |-  F  =  ( x  e.  A  |->  B )
43elrnmptg 4614 . 2  |-  ( A. x  e.  A  B  e.  _V  ->  ( C  e.  ran  F  <->  E. x  e.  A  C  =  B ) )
52, 4ax-mp 7 1  |-  ( C  e.  ran  F  <->  E. x  e.  A  C  =  B )
Colors of variables: wff set class
Syntax hints:    <-> wb 103    = wceq 1285    e. wcel 1434   A.wral 2349   E.wrex 2350   _Vcvv 2602    |-> cmpt 3847   ran crn 4372
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-mpt 3849  df-cnv 4379  df-dm 4381  df-rn 4382
This theorem is referenced by: (None)
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