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Theorem fvelrnb 5586
Description: A member of a function's range is a value of the function. (Contributed by NM, 31-Oct-1995.)
Assertion
Ref Expression
fvelrnb  |-  ( F  Fn  A  ->  ( B  e.  ran  F  <->  E. x  e.  A  ( F `  x )  =  B ) )
Distinct variable groups:    x, A    x, B    x, F

Proof of Theorem fvelrnb
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 fnrnfv 5585 . . 3  |-  ( F  Fn  A  ->  ran  F  =  { y  |  E. x  e.  A  y  =  ( F `  x ) } )
21eleq2d 2363 . 2  |-  ( F  Fn  A  ->  ( B  e.  ran  F  <->  B  e.  { y  |  E. x  e.  A  y  =  ( F `  x ) } ) )
3 fvex 5555 . . . . 5  |-  ( F `
 x )  e. 
_V
4 eleq1 2356 . . . . 5  |-  ( ( F `  x )  =  B  ->  (
( F `  x
)  e.  _V  <->  B  e.  _V ) )
53, 4mpbii 202 . . . 4  |-  ( ( F `  x )  =  B  ->  B  e.  _V )
65rexlimivw 2676 . . 3  |-  ( E. x  e.  A  ( F `  x )  =  B  ->  B  e.  _V )
7 eqeq1 2302 . . . . 5  |-  ( y  =  B  ->  (
y  =  ( F `
 x )  <->  B  =  ( F `  x ) ) )
8 eqcom 2298 . . . . 5  |-  ( B  =  ( F `  x )  <->  ( F `  x )  =  B )
97, 8syl6bb 252 . . . 4  |-  ( y  =  B  ->  (
y  =  ( F `
 x )  <->  ( F `  x )  =  B ) )
109rexbidv 2577 . . 3  |-  ( y  =  B  ->  ( E. x  e.  A  y  =  ( F `  x )  <->  E. x  e.  A  ( F `  x )  =  B ) )
116, 10elab3 2934 . 2  |-  ( B  e.  { y  |  E. x  e.  A  y  =  ( F `  x ) }  <->  E. x  e.  A  ( F `  x )  =  B )
122, 11syl6bb 252 1  |-  ( F  Fn  A  ->  ( B  e.  ran  F  <->  E. x  e.  A  ( F `  x )  =  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1632    e. wcel 1696   {cab 2282   E.wrex 2557   _Vcvv 2801   ran crn 4706    Fn wfn 5266   ` cfv 5271
This theorem is referenced by:  chfnrn  5652  rexrn  5683  ralrn  5684  ffnfv  5701  fconstfv  5750  elunirnALT  5795  isoini  5851  reldm  6187  canth  6310  seqomlem2  6479  fipreima  7177  ordiso2  7246  inf0  7338  inf3lem6  7350  noinfep  7376  noinfepOLD  7377  cantnflem4  7410  infenaleph  7734  isinfcard  7735  dfac5  7771  ackbij1  7880  sornom  7919  fin23lem16  7977  fin23lem21  7981  isf32lem2  7996  fin1a2lem5  8046  itunitc  8063  axdc3lem2  8093  zorn2lem4  8142  cfpwsdom  8222  peano2nn  9774  uzn0  10259  om2uzrani  11031  uzrdgfni  11037  uzin2  11844  unbenlem  12971  vdwlem6  13049  0ram  13083  imasmnd2  14425  imasgrp2  14626  pgpssslw  14941  efgsfo  15064  efgrelexlemb  15075  gexex  15161  imasrng  15418  2ndcomap  17200  kgenidm  17258  kqreglem1  17448  zfbas  17607  rnelfmlem  17663  rnelfm  17664  fmfnfmlem2  17666  ovolctb  18865  ovolicc2  18897  mbfinf  19036  dvivth  19373  dvne0  19374  mpfind  19444  mpfpf1  19450  pf1mpf  19451  aannenlem3  19726  reeff1o  19839  rnbra  22703  cnvbraval  22706  pjssdif1i  22771  dfpjop  22778  elpjrn  22786  ghomgrpilem2  24008  bwt2  25695  tailfb  26429  indexdom  26516  nacsfix  26890  lindfrn  27394  pmtrfrn  27503  fvelrnbf  27792  cncmpmax  27806  stoweidlem27  27879  stoweidlem31  27883  stoweidlem48  27900  stoweidlem59  27911  stirlinglem13  27938  usgraedgrn  28259  cdleme50rnlem  31355  diaelrnN  31857  diaintclN  31870  cdlemm10N  31930  dibintclN  31979  dihglb2  32154  dihintcl  32156  lcfrlem9  32362  mapd1o  32460  hdmaprnlem11N  32675  hgmaprnlem4N  32714
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-sbc 3005  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-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-iota 5235  df-fun 5273  df-fn 5274  df-fv 5279
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