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Theorem elrnmpti 4946
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 2623 . 2  |-  A. x  e.  A  B  e.  _V
3 rnmpt.1 . . 3  |-  F  =  ( x  e.  A  |->  B )
43elrnmptg 4945 . 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 1632    e. wcel 1696   A.wral 2556   E.wrex 2557   _Vcvv 2801    e. cmpt 4093   ran crn 4706
This theorem is referenced by:  fliftel  5824  oarec  6576  unfilem1  7137  pwfilem  7166  elrest  13348  iscyggen2  15184  iscyg3  15189  cycsubgcyg  15203  eldprd  15255  leordtval2  16958  iocpnfordt  16961  icomnfordt  16962  lecldbas  16965  tsmsxplem1  17851  minveclem2  18806  lhop2  19378  taylthlem2  19769  fsumvma  20468  dchrptlem2  20520  2sqlem1  20618  dchrisum0fno1  20676  minvecolem2  21470  esumcst  23451  esumpcvgval  23461  dya2iocct  23596  prsubrtr  25502  cntotbnd  26623  psgneldm2  27530  bnj1366  29178  islsat  29803
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-mpt 4095  df-cnv 4713  df-dm 4715  df-rn 4716
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