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Theorem elrnmpti 5105
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 2793 . 2  |-  A. x  e.  A  B  e.  _V
3 rnmpt.1 . . 3  |-  F  =  ( x  e.  A  |->  B )
43elrnmptg 5104 . 2  |-  ( A. x  e.  A  B  e.  _V  ->  ( C  e.  ran  F  <->  E. x  e.  A  C  =  B ) )
52, 4ax-mp 5 1  |-  ( C  e.  ran  F  <->  E. x  e.  A  C  =  B )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 187    = wceq 1437    e. wcel 1870   A.wral 2782   E.wrex 2783   _Vcvv 3087    |-> cmpt 4484   ran crn 4855
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pr 4661
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-br 4427  df-opab 4485  df-mpt 4486  df-cnv 4862  df-dm 4864  df-rn 4865
This theorem is referenced by:  fliftel  6217  oarec  7271  unfilem1  7841  pwfilem  7874  elrest  15289  psgneldm2  17100  psgnfitr  17113  iscyggen2  17455  iscyg3  17460  cycsubgcyg  17474  eldprd  17575  leordtval2  20163  iocpnfordt  20166  icomnfordt  20167  lecldbas  20170  tsmsxplem1  21102  minveclem2  22265  lhop2  22852  taylthlem2  23202  fsumvma  24012  dchrptlem2  24064  2sqlem1  24162  dchrisum0fno1  24220  minvecolem2  26370  gsumesum  28727  esumlub  28728  esumcst  28731  esumpcvgval  28746  esumgect  28758  esum2d  28761  sigapildsys  28831  sxbrsigalem2  28955  omssubaddlem  28968  omssubadd  28969  eulerpartgbij  29039  bnj1366  29437  msubco  29965  msubvrs  29994  fin2so  31647  poimirlem17  31672  poimirlem20  31675  cntotbnd  31843  islsat  32277
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