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Theorem fnniniseg2 5591
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 5589 . 2  |-  ( F  Fn  A  ->  ( `' F " ( _V 
\  { B }
) )  =  {
x  e.  A  | 
( F `  x
)  e.  ( _V 
\  { B }
) } )
2 eldifsn 3687 . . . 4  |-  ( ( F `  x )  e.  ( _V  \  { B } )  <->  ( ( F `  x )  e.  _V  /\  ( F `
 x )  =/= 
B ) )
3 funfvex 5486 . . . . . 6  |-  ( ( Fun  F  /\  x  e.  dom  F )  -> 
( F `  x
)  e.  _V )
43funfni 5271 . . . . 5  |-  ( ( F  Fn  A  /\  x  e.  A )  ->  ( F `  x
)  e.  _V )
54biantrurd 303 . . . 4  |-  ( ( F  Fn  A  /\  x  e.  A )  ->  ( ( F `  x )  =/=  B  <->  ( ( F `  x
)  e.  _V  /\  ( F `  x )  =/=  B ) ) )
62, 5bitr4id 198 . . 3  |-  ( ( F  Fn  A  /\  x  e.  A )  ->  ( ( F `  x )  e.  ( _V  \  { B } )  <->  ( F `  x )  =/=  B
) )
76rabbidva 2700 . 2  |-  ( F  Fn  A  ->  { x  e.  A  |  ( F `  x )  e.  ( _V  \  { B } ) }  =  { x  e.  A  |  ( F `  x )  =/=  B } )
81, 7eqtrd 2190 1  |-  ( F  Fn  A  ->  ( `' F " ( _V 
\  { B }
) )  =  {
x  e.  A  | 
( F `  x
)  =/=  B }
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1335    e. wcel 2128    =/= wne 2327   {crab 2439   _Vcvv 2712    \ cdif 3099   {csn 3560   `'ccnv 4586   "cima 4590    Fn wfn 5166   ` cfv 5171
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4083  ax-pow 4136  ax-pr 4170
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-rab 2444  df-v 2714  df-sbc 2938  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3774  df-br 3967  df-opab 4027  df-id 4254  df-xp 4593  df-rel 4594  df-cnv 4595  df-co 4596  df-dm 4597  df-rn 4598  df-res 4599  df-ima 4600  df-iota 5136  df-fun 5173  df-fn 5174  df-fv 5179
This theorem is referenced by: (None)
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