MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  elrnmpti Structured version   Unicode version

Theorem elrnmpti 5122
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 2774 . 2  |-  A. x  e.  A  B  e.  _V
3 rnmpt.1 . . 3  |-  F  =  ( x  e.  A  |->  B )
43elrnmptg 5121 . 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 178    = wceq 1653    e. wcel 1726   A.wral 2706   E.wrex 2707   _Vcvv 2957    e. cmpt 4267   ran crn 4880
This theorem is referenced by:  fliftel  6032  oarec  6806  unfilem1  7372  pwfilem  7402  elrest  13656  iscyggen2  15492  iscyg3  15497  cycsubgcyg  15511  eldprd  15563  leordtval2  17277  iocpnfordt  17280  icomnfordt  17281  lecldbas  17284  tsmsxplem1  18183  minveclem2  19328  lhop2  19900  taylthlem2  20291  fsumvma  20998  dchrptlem2  21050  2sqlem1  21148  dchrisum0fno1  21206  minvecolem2  22378  gsumesum  24452  esumlub  24453  esumcst  24456  esumpcvgval  24469  sxbrsigalem2  24637  cntotbnd  26506  psgneldm2  27405  bnj1366  29202  islsat  29790
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pr 4404
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-op 3824  df-br 4214  df-opab 4268  df-mpt 4269  df-cnv 4887  df-dm 4889  df-rn 4890
  Copyright terms: Public domain W3C validator