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Theorem rexrn 5667
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 5539 . . 3  |-  ( F `
 y )  e. 
_V
21a1i 10 . 2  |-  ( ( F  Fn  A  /\  y  e.  A )  ->  ( F `  y
)  e.  _V )
3 fvelrnb 5570 . . 3  |-  ( F  Fn  A  ->  (
x  e.  ran  F  <->  E. y  e.  A  ( F `  y )  =  x ) )
4 eqcom 2285 . . . 4  |-  ( ( F `  y )  =  x  <->  x  =  ( F `  y ) )
54rexbii 2568 . . 3  |-  ( E. y  e.  A  ( F `  y )  =  x  <->  E. y  e.  A  x  =  ( F `  y ) )
63, 5syl6bb 252 . 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 452 . 2  |-  ( ( F  Fn  A  /\  x  =  ( F `  y ) )  -> 
( ph  <->  ps ) )
92, 6, 8rexxfr2d 4551 1  |-  ( F  Fn  A  ->  ( E. x  e.  ran  F
ph 
<->  E. y  e.  A  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   E.wrex 2544   _Vcvv 2788   ran crn 4690    Fn wfn 5250   ` cfv 5255
This theorem is referenced by:  wemapwe  7400  rexanuz  11829  climsup  12143  supcvg  12314  ruclem12  12519  prmreclem6  12968  vdwmc  13025  znunit  16517  lmbr2  16989  lmff  17029  1stcfb  17171  imasf1oxms  18035  lebnumlem3  18461  lmmbr2  18685  lmcau  18738  bcthlem4  18749  mbfsup  19019  itg2monolem1  19105  itg2gt0  19115  ostth  20788  erdszelem10  23731  neibastop2lem  26309  filnetlem4  26330  istotbnd3  26495  sstotbnd  26499  heibor  26545  nacsfix  26787  fnwe2lem2  27148  climinf  27732
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fn 5258  df-fv 5263
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