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Theorem fnniniseg2 5342
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 5340 . 2  |-  ( F  Fn  A  ->  ( `' F " ( _V 
\  { B }
) )  =  {
x  e.  A  | 
( F `  x
)  e.  ( _V 
\  { B }
) } )
2 funfvex 5243 . . . . . 6  |-  ( ( Fun  F  /\  x  e.  dom  F )  -> 
( F `  x
)  e.  _V )
32funfni 5050 . . . . 5  |-  ( ( F  Fn  A  /\  x  e.  A )  ->  ( F `  x
)  e.  _V )
43biantrurd 299 . . . 4  |-  ( ( F  Fn  A  /\  x  e.  A )  ->  ( ( F `  x )  =/=  B  <->  ( ( F `  x
)  e.  _V  /\  ( F `  x )  =/=  B ) ) )
5 eldifsn 3535 . . . 4  |-  ( ( F `  x )  e.  ( _V  \  { B } )  <->  ( ( F `  x )  e.  _V  /\  ( F `
 x )  =/= 
B ) )
64, 5syl6rbbr 197 . . 3  |-  ( ( F  Fn  A  /\  x  e.  A )  ->  ( ( F `  x )  e.  ( _V  \  { B } )  <->  ( F `  x )  =/=  B
) )
76rabbidva 2598 . 2  |-  ( F  Fn  A  ->  { x  e.  A  |  ( F `  x )  e.  ( _V  \  { B } ) }  =  { x  e.  A  |  ( F `  x )  =/=  B } )
81, 7eqtrd 2115 1  |-  ( F  Fn  A  ->  ( `' F " ( _V 
\  { B }
) )  =  {
x  e.  A  | 
( F `  x
)  =/=  B }
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1285    e. wcel 1434    =/= wne 2249   {crab 2357   _Vcvv 2610    \ cdif 2979   {csn 3416   `'ccnv 4390   "cima 4394    Fn wfn 4947   ` cfv 4952
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 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-rab 2362  df-v 2612  df-sbc 2825  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-br 3806  df-opab 3860  df-id 4076  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-res 4403  df-ima 4404  df-iota 4917  df-fun 4954  df-fn 4955  df-fv 4960
This theorem is referenced by: (None)
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