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Theorem rexrn 6009
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 5858 . . 3  |-  ( F `
 y )  e. 
_V
21a1i 11 . 2  |-  ( ( F  Fn  A  /\  y  e.  A )  ->  ( F `  y
)  e.  _V )
3 fvelrnb 5895 . . 3  |-  ( F  Fn  A  ->  (
x  e.  ran  F  <->  E. y  e.  A  ( F `  y )  =  x ) )
4 eqcom 2463 . . . 4  |-  ( ( F `  y )  =  x  <->  x  =  ( F `  y ) )
54rexbii 2956 . . 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 464 . 2  |-  ( ( F  Fn  A  /\  x  =  ( F `  y ) )  -> 
( ph  <->  ps ) )
92, 6, 8rexxfr2d 4654 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 367    = wceq 1398    e. wcel 1823   E.wrex 2805   _Vcvv 3106   ran crn 4989    Fn wfn 5565   ` cfv 5570
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-iota 5534  df-fun 5572  df-fn 5573  df-fv 5578
This theorem is referenced by:  elrnrexdm  6011  wemapwe  8130  wemapweOLD  8131  rexanuz  13263  climsup  13577  supcvg  13752  ruclem12  14061  prmreclem6  14526  vdwmc  14583  znunit  18778  lmbr2  19930  lmff  19972  1stcfb  20115  imasf1oxms  21161  lebnumlem3  21632  lmmbr2  21867  lmcau  21920  bcthlem4  21935  mbfsup  22240  itg2monolem1  22326  itg2gt0  22336  ostth  24025  erdszelem10  28911  mblfinlem2  30295  neibastop2lem  30421  filnetlem4  30442  istotbnd3  30510  sstotbnd  30514  heibor  30560  nacsfix  30887  fnwe2lem2  31239  climinf  31854
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