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Theorem fnniniseg2 5687
Description: Support sets of functions expressed as abstractions. (Contributed by Stefan O'Rear, 1-Feb-2015.)
Assertion
Ref Expression
fnniniseg2  |-  ( F  Fn  A  ->  ( `' F " ( _V 
\  { B }
) )  =  {
x  e.  A  | 
( F `  x
)  =/=  B }
)
Distinct variable groups:    x, A    x, F    x, B

Proof of Theorem fnniniseg2
StepHypRef Expression
1 fncnvima2 5685 . 2  |-  ( F  Fn  A  ->  ( `' F " ( _V 
\  { B }
) )  =  {
x  e.  A  | 
( F `  x
)  e.  ( _V 
\  { B }
) } )
2 fvex 5577 . . . . 5  |-  ( F `
 x )  e. 
_V
3 eldifsn 3783 . . . . 5  |-  ( ( F `  x )  e.  ( _V  \  { B } )  <->  ( ( F `  x )  e.  _V  /\  ( F `
 x )  =/= 
B ) )
42, 3mpbiran 884 . . . 4  |-  ( ( F `  x )  e.  ( _V  \  { B } )  <->  ( F `  x )  =/=  B
)
54a1i 10 . . 3  |-  ( x  e.  A  ->  (
( F `  x
)  e.  ( _V 
\  { B }
)  <->  ( F `  x )  =/=  B
) )
65rabbiia 2812 . 2  |-  { x  e.  A  |  ( F `  x )  e.  ( _V  \  { B } ) }  =  { x  e.  A  |  ( F `  x )  =/=  B }
71, 6syl6eq 2364 1  |-  ( F  Fn  A  ->  ( `' F " ( _V 
\  { B }
) )  =  {
x  e.  A  | 
( F `  x
)  =/=  B }
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1633    e. wcel 1701    =/= wne 2479   {crab 2581   _Vcvv 2822    \ cdif 3183   {csn 3674   `'ccnv 4725   "cima 4729    Fn wfn 5287   ` cfv 5292
This theorem is referenced by:  fnsuppres  5773  frlmbas  26371  frlmssuvc2  26395
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pr 4251
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-rab 2586  df-v 2824  df-sbc 3026  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-br 4061  df-opab 4115  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-fv 5300
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