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Theorem elrnmpti 4930
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 2610 . 2  |-  A. x  e.  A  B  e.  _V
3 rnmpt.1 . . 3  |-  F  =  ( x  e.  A  |->  B )
43elrnmptg 4929 . 2  |-  ( A. x  e.  A  B  e.  _V  ->  ( C  e.  ran  F  <->  E. x  e.  A  C  =  B ) )
52, 4ax-mp 8 1  |-  ( C  e.  ran  F  <->  E. x  e.  A  C  =  B )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544   _Vcvv 2788    e. cmpt 4077   ran crn 4690
This theorem is referenced by:  fliftel  5808  oarec  6560  unfilem1  7121  pwfilem  7150  elrest  13332  iscyggen2  15168  iscyg3  15173  cycsubgcyg  15187  eldprd  15239  leordtval2  16942  iocpnfordt  16945  icomnfordt  16946  lecldbas  16949  tsmsxplem1  17835  minveclem2  18790  lhop2  19362  taylthlem2  19753  fsumvma  20452  dchrptlem2  20504  2sqlem1  20602  dchrisum0fno1  20660  minvecolem2  21454  esumcst  23436  esumpcvgval  23446  dya2iocct  23581  prsubrtr  25399  cntotbnd  26520  psgneldm2  27427  bnj1366  28862  islsat  29181
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-mpt 4079  df-cnv 4697  df-dm 4699  df-rn 4700
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