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Theorem rexrn 5872
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 5742 . . 3  |-  ( F `
 y )  e. 
_V
21a1i 11 . 2  |-  ( ( F  Fn  A  /\  y  e.  A )  ->  ( F `  y
)  e.  _V )
3 fvelrnb 5774 . . 3  |-  ( F  Fn  A  ->  (
x  e.  ran  F  <->  E. y  e.  A  ( F `  y )  =  x ) )
4 eqcom 2438 . . . 4  |-  ( ( F `  y )  =  x  <->  x  =  ( F `  y ) )
54rexbii 2730 . . 3  |-  ( E. y  e.  A  ( F `  y )  =  x  <->  E. y  e.  A  x  =  ( F `  y ) )
63, 5syl6bb 253 . 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 453 . 2  |-  ( ( F  Fn  A  /\  x  =  ( F `  y ) )  -> 
( ph  <->  ps ) )
92, 6, 8rexxfr2d 4740 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 177    /\ wa 359    = wceq 1652    e. wcel 1725   E.wrex 2706   _Vcvv 2956   ran crn 4879    Fn wfn 5449   ` cfv 5454
This theorem is referenced by:  elrnrexdm  5874  wemapwe  7654  rexanuz  12149  climsup  12463  supcvg  12635  ruclem12  12840  prmreclem6  13289  vdwmc  13346  znunit  16844  lmbr2  17323  lmff  17365  1stcfb  17508  imasf1oxms  18519  lebnumlem3  18988  lmmbr2  19212  lmcau  19265  bcthlem4  19280  mbfsup  19556  itg2monolem1  19642  itg2gt0  19652  ostth  21333  erdszelem10  24886  mblfinlem2  26244  neibastop2lem  26389  filnetlem4  26410  istotbnd3  26480  sstotbnd  26484  heibor  26530  nacsfix  26766  fnwe2lem2  27126  climinf  27708
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-iota 5418  df-fun 5456  df-fn 5457  df-fv 5462
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