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Theorem fnniniseg2 5290
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 5288 . 2  |-  ( F  Fn  A  ->  ( `' F " ( _V 
\  { B }
) )  =  {
x  e.  A  | 
( F `  x
)  e.  ( _V 
\  { B }
) } )
2 funfvex 5192 . . . . . 6  |-  ( ( Fun  F  /\  x  e.  dom  F )  -> 
( F `  x
)  e.  _V )
32funfni 4999 . . . . 5  |-  ( ( F  Fn  A  /\  x  e.  A )  ->  ( F `  x
)  e.  _V )
43biantrurd 289 . . . 4  |-  ( ( F  Fn  A  /\  x  e.  A )  ->  ( ( F `  x )  =/=  B  <->  ( ( F `  x
)  e.  _V  /\  ( F `  x )  =/=  B ) ) )
5 eldifsn 3495 . . . 4  |-  ( ( F `  x )  e.  ( _V  \  { B } )  <->  ( ( F `  x )  e.  _V  /\  ( F `
 x )  =/= 
B ) )
64, 5syl6rbbr 188 . . 3  |-  ( ( F  Fn  A  /\  x  e.  A )  ->  ( ( F `  x )  e.  ( _V  \  { B } )  <->  ( F `  x )  =/=  B
) )
76rabbidva 2548 . 2  |-  ( F  Fn  A  ->  { x  e.  A  |  ( F `  x )  e.  ( _V  \  { B } ) }  =  { x  e.  A  |  ( F `  x )  =/=  B } )
81, 7eqtrd 2072 1  |-  ( F  Fn  A  ->  ( `' F " ( _V 
\  { B }
) )  =  {
x  e.  A  | 
( F `  x
)  =/=  B }
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 97    = wceq 1243    e. wcel 1393    =/= wne 2204   {crab 2310   _Vcvv 2557    \ cdif 2914   {csn 3375   `'ccnv 4344   "cima 4348    Fn wfn 4897   ` cfv 4902
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-rab 2315  df-v 2559  df-sbc 2765  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-fv 4910
This theorem is referenced by: (None)
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