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Theorem rexsupp 5847
Description: Existential quantification restricted to a support. (Contributed by Stefan O'Rear, 23-Mar-2015.)
Assertion
Ref Expression
rexsupp  |-  ( F  Fn  A  ->  ( E. x  e.  ( `' F " ( _V 
\  { Z }
) ) ph  <->  E. x  e.  A  ( ( F `  x )  =/=  Z  /\  ph )
) )
Distinct variable groups:    x, F    x, A
Allowed substitution hints:    ph( x)    Z( x)

Proof of Theorem rexsupp
StepHypRef Expression
1 elpreima 5842 . . . . 5  |-  ( F  Fn  A  ->  (
x  e.  ( `' F " ( _V 
\  { Z }
) )  <->  ( x  e.  A  /\  ( F `  x )  e.  ( _V  \  { Z } ) ) ) )
2 fvex 5734 . . . . . . 7  |-  ( F `
 x )  e. 
_V
3 eldifsn 3919 . . . . . . 7  |-  ( ( F `  x )  e.  ( _V  \  { Z } )  <->  ( ( F `  x )  e.  _V  /\  ( F `
 x )  =/= 
Z ) )
42, 3mpbiran 885 . . . . . 6  |-  ( ( F `  x )  e.  ( _V  \  { Z } )  <->  ( F `  x )  =/=  Z
)
54anbi2i 676 . . . . 5  |-  ( ( x  e.  A  /\  ( F `  x )  e.  ( _V  \  { Z } ) )  <-> 
( x  e.  A  /\  ( F `  x
)  =/=  Z ) )
61, 5syl6bb 253 . . . 4  |-  ( F  Fn  A  ->  (
x  e.  ( `' F " ( _V 
\  { Z }
) )  <->  ( x  e.  A  /\  ( F `  x )  =/=  Z ) ) )
76anbi1d 686 . . 3  |-  ( F  Fn  A  ->  (
( x  e.  ( `' F " ( _V 
\  { Z }
) )  /\  ph ) 
<->  ( ( x  e.  A  /\  ( F `
 x )  =/= 
Z )  /\  ph ) ) )
8 anass 631 . . 3  |-  ( ( ( x  e.  A  /\  ( F `  x
)  =/=  Z )  /\  ph )  <->  ( x  e.  A  /\  (
( F `  x
)  =/=  Z  /\  ph ) ) )
97, 8syl6bb 253 . 2  |-  ( F  Fn  A  ->  (
( x  e.  ( `' F " ( _V 
\  { Z }
) )  /\  ph ) 
<->  ( x  e.  A  /\  ( ( F `  x )  =/=  Z  /\  ph ) ) ) )
109rexbidv2 2720 1  |-  ( F  Fn  A  ->  ( E. x  e.  ( `' F " ( _V 
\  { Z }
) ) ph  <->  E. x  e.  A  ( ( F `  x )  =/=  Z  /\  ph )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    e. wcel 1725    =/= wne 2598   E.wrex 2698   _Vcvv 2948    \ cdif 3309   {csn 3806   `'ccnv 4869   "cima 4873    Fn wfn 5441   ` cfv 5446
This theorem is referenced by:  mdegldg  19981
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 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-fv 5454
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