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Theorem rexsupp 5371
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 5366 . . . . 5  |-  ( F  Fn  A  ->  (
x  e.  ( `' F " ( _V 
\  { Z }
) )  <->  ( x  e.  A  /\  ( F `  x )  e.  ( _V  \  { Z } ) ) ) )
2 funfvex 5270 . . . . . . . . 9  |-  ( ( Fun  F  /\  x  e.  dom  F )  -> 
( F `  x
)  e.  _V )
32funfni 5070 . . . . . . . 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 3544 . . . . . . 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 2379 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 1436    =/= wne 2251   E.wrex 2356   _Vcvv 2614    \ cdif 2983   {csn 3425   `'ccnv 4403   "cima 4407    Fn wfn 4967   ` cfv 4972
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 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3925  ax-pow 3977  ax-pr 4003
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ne 2252  df-ral 2360  df-rex 2361  df-v 2616  df-sbc 2829  df-dif 2988  df-un 2990  df-in 2992  df-ss 2999  df-pw 3411  df-sn 3431  df-pr 3432  df-op 3434  df-uni 3631  df-br 3815  df-opab 3869  df-id 4087  df-xp 4410  df-rel 4411  df-cnv 4412  df-co 4413  df-dm 4414  df-rn 4415  df-res 4416  df-ima 4417  df-iota 4937  df-fun 4974  df-fn 4975  df-fv 4980
This theorem is referenced by: (None)
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