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Theorem elrnmpti 5253
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 2825 . 2  |-  A. x  e.  A  B  e.  _V
3 rnmpt.1 . . 3  |-  F  =  ( x  e.  A  |->  B )
43elrnmptg 5252 . 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 184    = wceq 1379    e. wcel 1767   A.wral 2814   E.wrex 2815   _Vcvv 3113    |-> cmpt 4505   ran crn 5000
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-opab 4506  df-mpt 4507  df-cnv 5007  df-dm 5009  df-rn 5010
This theorem is referenced by:  fliftel  6195  oarec  7211  unfilem1  7784  pwfilem  7814  elrest  14683  psgneldm2  16335  psgnfitr  16348  iscyggen2  16687  iscyg3  16692  cycsubgcyg  16706  eldprd  16838  eldprdOLD  16840  leordtval2  19507  iocpnfordt  19510  icomnfordt  19511  lecldbas  19514  tsmsxplem1  20418  minveclem2  21604  lhop2  22179  taylthlem2  22531  fsumvma  23244  dchrptlem2  23296  2sqlem1  23394  dchrisum0fno1  23452  minvecolem2  25495  gsumesum  27735  esumlub  27736  esumcst  27739  esumpcvgval  27752  sxbrsigalem2  27925  eulerpartgbij  27979  fin2so  29645  cntotbnd  29923  bnj1366  32985  islsat  33806
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