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

Theorem elrnmpti 5089
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 2782 . 2  |-  A. x  e.  A  B  e.  _V
3 rnmpt.1 . . 3  |-  F  =  ( x  e.  A  |->  B )
43elrnmptg 5088 . 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 1369    e. wcel 1756   A.wral 2714   E.wrex 2715   _Vcvv 2971    e. cmpt 4349   ran crn 4840
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4412  ax-nul 4420  ax-pr 4530
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2719  df-rex 2720  df-rab 2723  df-v 2973  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-nul 3637  df-if 3791  df-sn 3877  df-pr 3879  df-op 3883  df-br 4292  df-opab 4350  df-mpt 4351  df-cnv 4847  df-dm 4849  df-rn 4850
This theorem is referenced by:  fliftel  6001  oarec  7000  unfilem1  7575  pwfilem  7604  elrest  14365  psgneldm2  16009  psgnfitr  16022  iscyggen2  16357  iscyg3  16362  cycsubgcyg  16376  eldprd  16485  eldprdOLD  16487  leordtval2  18815  iocpnfordt  18818  icomnfordt  18819  lecldbas  18822  tsmsxplem1  19726  minveclem2  20912  lhop2  21486  taylthlem2  21838  fsumvma  22551  dchrptlem2  22603  2sqlem1  22701  dchrisum0fno1  22759  minvecolem2  24275  gsumesum  26509  esumlub  26510  esumcst  26513  esumpcvgval  26526  sxbrsigalem2  26700  eulerpartgbij  26754  fin2so  28414  cntotbnd  28693  bnj1366  31821  islsat  32634
  Copyright terms: Public domain W3C validator