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Theorem rexsupp 5661
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 5656 . . . . 5  |-  ( F  Fn  A  ->  (
x  e.  ( `' F " ( _V 
\  { Z }
) )  <->  ( x  e.  A  /\  ( F `  x )  e.  ( _V  \  { Z } ) ) ) )
2 eldifsn 3734 . . . . . . 7  |-  ( ( F `  x )  e.  ( _V  \  { Z } )  <->  ( ( F `  x )  e.  _V  /\  ( F `
 x )  =/= 
Z ) )
3 funfvex 5551 . . . . . . . . 9  |-  ( ( Fun  F  /\  x  e.  dom  F )  -> 
( F `  x
)  e.  _V )
43funfni 5335 . . . . . . . 8  |-  ( ( F  Fn  A  /\  x  e.  A )  ->  ( F `  x
)  e.  _V )
54biantrurd 305 . . . . . . 7  |-  ( ( F  Fn  A  /\  x  e.  A )  ->  ( ( F `  x )  =/=  Z  <->  ( ( F `  x
)  e.  _V  /\  ( F `  x )  =/=  Z ) ) )
62, 5bitr4id 199 . . . . . 6  |-  ( ( F  Fn  A  /\  x  e.  A )  ->  ( ( F `  x )  e.  ( _V  \  { Z } )  <->  ( F `  x )  =/=  Z
) )
76pm5.32da 452 . . . . 5  |-  ( F  Fn  A  ->  (
( x  e.  A  /\  ( F `  x
)  e.  ( _V 
\  { Z }
) )  <->  ( x  e.  A  /\  ( F `  x )  =/=  Z ) ) )
81, 7bitrd 188 . . . 4  |-  ( F  Fn  A  ->  (
x  e.  ( `' F " ( _V 
\  { Z }
) )  <->  ( x  e.  A  /\  ( F `  x )  =/=  Z ) ) )
98anbi1d 465 . . 3  |-  ( F  Fn  A  ->  (
( x  e.  ( `' F " ( _V 
\  { Z }
) )  /\  ph ) 
<->  ( ( x  e.  A  /\  ( F `
 x )  =/= 
Z )  /\  ph ) ) )
10 anass 401 . . 3  |-  ( ( ( x  e.  A  /\  ( F `  x
)  =/=  Z )  /\  ph )  <->  ( x  e.  A  /\  (
( F `  x
)  =/=  Z  /\  ph ) ) )
119, 10bitrdi 196 . 2  |-  ( F  Fn  A  ->  (
( x  e.  ( `' F " ( _V 
\  { Z }
) )  /\  ph ) 
<->  ( x  e.  A  /\  ( ( F `  x )  =/=  Z  /\  ph ) ) ) )
1211rexbidv2 2493 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 104    <-> wb 105    e. wcel 2160    =/= wne 2360   E.wrex 2469   _Vcvv 2752    \ cdif 3141   {csn 3607   `'ccnv 4643   "cima 4647    Fn wfn 5230   ` cfv 5235
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-v 2754  df-sbc 2978  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-fv 5243
This theorem is referenced by: (None)
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