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Theorem rexsupp 5423
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 5418 . . . . 5  |-  ( F  Fn  A  ->  (
x  e.  ( `' F " ( _V 
\  { Z }
) )  <->  ( x  e.  A  /\  ( F `  x )  e.  ( _V  \  { Z } ) ) ) )
2 funfvex 5322 . . . . . . . . 9  |-  ( ( Fun  F  /\  x  e.  dom  F )  -> 
( F `  x
)  e.  _V )
32funfni 5114 . . . . . . . 8  |-  ( ( F  Fn  A  /\  x  e.  A )  ->  ( F `  x
)  e.  _V )
43biantrurd 299 . . . . . . 7  |-  ( ( F  Fn  A  /\  x  e.  A )  ->  ( ( F `  x )  =/=  Z  <->  ( ( F `  x
)  e.  _V  /\  ( F `  x )  =/=  Z ) ) )
5 eldifsn 3567 . . . . . . 7  |-  ( ( F `  x )  e.  ( _V  \  { Z } )  <->  ( ( F `  x )  e.  _V  /\  ( F `
 x )  =/= 
Z ) )
64, 5syl6rbbr 197 . . . . . 6  |-  ( ( F  Fn  A  /\  x  e.  A )  ->  ( ( F `  x )  e.  ( _V  \  { Z } )  <->  ( F `  x )  =/=  Z
) )
76pm5.32da 440 . . . . 5  |-  ( F  Fn  A  ->  (
( x  e.  A  /\  ( F `  x
)  e.  ( _V 
\  { Z }
) )  <->  ( x  e.  A  /\  ( F `  x )  =/=  Z ) ) )
81, 7bitrd 186 . . . 4  |-  ( F  Fn  A  ->  (
x  e.  ( `' F " ( _V 
\  { Z }
) )  <->  ( x  e.  A  /\  ( F `  x )  =/=  Z ) ) )
98anbi1d 453 . . 3  |-  ( F  Fn  A  ->  (
( x  e.  ( `' F " ( _V 
\  { Z }
) )  /\  ph ) 
<->  ( ( x  e.  A  /\  ( F `
 x )  =/= 
Z )  /\  ph ) ) )
10 anass 393 . . 3  |-  ( ( ( x  e.  A  /\  ( F `  x
)  =/=  Z )  /\  ph )  <->  ( x  e.  A  /\  (
( F `  x
)  =/=  Z  /\  ph ) ) )
119, 10syl6bb 194 . 2  |-  ( F  Fn  A  ->  (
( x  e.  ( `' F " ( _V 
\  { Z }
) )  /\  ph ) 
<->  ( x  e.  A  /\  ( ( F `  x )  =/=  Z  /\  ph ) ) ) )
1211rexbidv2 2383 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 102    <-> wb 103    e. wcel 1438    =/= wne 2255   E.wrex 2360   _Vcvv 2619    \ cdif 2996   {csn 3446   `'ccnv 4437   "cima 4441    Fn wfn 5010   ` cfv 5015
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2841  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-opab 3900  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-fv 5023
This theorem is referenced by: (None)
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