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Theorem rexrn 5845
Description: Restricted existential quantification over the range of a function. (Contributed by Mario Carneiro, 24-Dec-2013.) (Revised by Mario Carneiro, 20-Aug-2014.)
Hypothesis
Ref Expression
rexrn.1  |-  ( x  =  ( F `  y )  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
rexrn  |-  ( F  Fn  A  ->  ( E. x  e.  ran  F
ph 
<->  E. y  e.  A  ps ) )
Distinct variable groups:    x, y, A    x, F, y    ps, x    ph, y
Allowed substitution hints:    ph( x)    ps( y)

Proof of Theorem rexrn
StepHypRef Expression
1 fvex 5701 . . 3  |-  ( F `
 y )  e. 
_V
21a1i 11 . 2  |-  ( ( F  Fn  A  /\  y  e.  A )  ->  ( F `  y
)  e.  _V )
3 fvelrnb 5739 . . 3  |-  ( F  Fn  A  ->  (
x  e.  ran  F  <->  E. y  e.  A  ( F `  y )  =  x ) )
4 eqcom 2445 . . . 4  |-  ( ( F `  y )  =  x  <->  x  =  ( F `  y ) )
54rexbii 2740 . . 3  |-  ( E. y  e.  A  ( F `  y )  =  x  <->  E. y  e.  A  x  =  ( F `  y ) )
63, 5syl6bb 261 . 2  |-  ( F  Fn  A  ->  (
x  e.  ran  F  <->  E. y  e.  A  x  =  ( F `  y ) ) )
7 rexrn.1 . . 3  |-  ( x  =  ( F `  y )  ->  ( ph 
<->  ps ) )
87adantl 466 . 2  |-  ( ( F  Fn  A  /\  x  =  ( F `  y ) )  -> 
( ph  <->  ps ) )
92, 6, 8rexxfr2d 4509 1  |-  ( F  Fn  A  ->  ( E. x  e.  ran  F
ph 
<->  E. y  e.  A  ps ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   E.wrex 2716   _Vcvv 2972   ran crn 4841    Fn wfn 5413   ` cfv 5418
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 4413  ax-nul 4421  ax-pr 4531
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 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-sbc 3187  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-iota 5381  df-fun 5420  df-fn 5421  df-fv 5426
This theorem is referenced by:  elrnrexdm  5847  wemapwe  7928  wemapweOLD  7929  rexanuz  12833  climsup  13147  supcvg  13318  ruclem12  13523  prmreclem6  13982  vdwmc  14039  znunit  17996  lmbr2  18863  lmff  18905  1stcfb  19049  imasf1oxms  20064  lebnumlem3  20535  lmmbr2  20770  lmcau  20823  bcthlem4  20838  mbfsup  21142  itg2monolem1  21228  itg2gt0  21238  ostth  22888  erdszelem10  27088  mblfinlem2  28429  neibastop2lem  28581  filnetlem4  28602  istotbnd3  28670  sstotbnd  28674  heibor  28720  nacsfix  29048  fnwe2lem2  29404  climinf  29779
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