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Theorem rexsupp 5537
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 5532 . . . . 5  |-  ( F  Fn  A  ->  (
x  e.  ( `' F " ( _V 
\  { Z }
) )  <->  ( x  e.  A  /\  ( F `  x )  e.  ( _V  \  { Z } ) ) ) )
2 funfvex 5431 . . . . . . . . 9  |-  ( ( Fun  F  /\  x  e.  dom  F )  -> 
( F `  x
)  e.  _V )
32funfni 5218 . . . . . . . 8  |-  ( ( F  Fn  A  /\  x  e.  A )  ->  ( F `  x
)  e.  _V )
43biantrurd 303 . . . . . . 7  |-  ( ( F  Fn  A  /\  x  e.  A )  ->  ( ( F `  x )  =/=  Z  <->  ( ( F `  x
)  e.  _V  /\  ( F `  x )  =/=  Z ) ) )
5 eldifsn 3645 . . . . . . 7  |-  ( ( F `  x )  e.  ( _V  \  { Z } )  <->  ( ( F `  x )  e.  _V  /\  ( F `
 x )  =/= 
Z ) )
64, 5syl6rbbr 198 . . . . . 6  |-  ( ( F  Fn  A  /\  x  e.  A )  ->  ( ( F `  x )  e.  ( _V  \  { Z } )  <->  ( F `  x )  =/=  Z
) )
76pm5.32da 447 . . . . 5  |-  ( F  Fn  A  ->  (
( x  e.  A  /\  ( F `  x
)  e.  ( _V 
\  { Z }
) )  <->  ( x  e.  A  /\  ( F `  x )  =/=  Z ) ) )
81, 7bitrd 187 . . . 4  |-  ( F  Fn  A  ->  (
x  e.  ( `' F " ( _V 
\  { Z }
) )  <->  ( x  e.  A  /\  ( F `  x )  =/=  Z ) ) )
98anbi1d 460 . . 3  |-  ( F  Fn  A  ->  (
( x  e.  ( `' F " ( _V 
\  { Z }
) )  /\  ph ) 
<->  ( ( x  e.  A  /\  ( F `
 x )  =/= 
Z )  /\  ph ) ) )
10 anass 398 . . 3  |-  ( ( ( x  e.  A  /\  ( F `  x
)  =/=  Z )  /\  ph )  <->  ( x  e.  A  /\  (
( F `  x
)  =/=  Z  /\  ph ) ) )
119, 10syl6bb 195 . 2  |-  ( F  Fn  A  ->  (
( x  e.  ( `' F " ( _V 
\  { Z }
) )  /\  ph ) 
<->  ( x  e.  A  /\  ( ( F `  x )  =/=  Z  /\  ph ) ) ) )
1211rexbidv2 2438 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    /\ wa 103    <-> wb 104    e. wcel 1480    =/= wne 2306   E.wrex 2415   _Vcvv 2681    \ cdif 3063   {csn 3522   `'ccnv 4533   "cima 4537    Fn wfn 5113   ` cfv 5118
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-ral 2419  df-rex 2420  df-v 2683  df-sbc 2905  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-fv 5126
This theorem is referenced by: (None)
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