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Theorem fnniniseg2 5619
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 5617 . 2  |-  ( F  Fn  A  ->  ( `' F " ( _V 
\  { B }
) )  =  {
x  e.  A  | 
( F `  x
)  e.  ( _V 
\  { B }
) } )
2 eldifsn 3710 . . . 4  |-  ( ( F `  x )  e.  ( _V  \  { B } )  <->  ( ( F `  x )  e.  _V  /\  ( F `
 x )  =/= 
B ) )
3 funfvex 5513 . . . . . 6  |-  ( ( Fun  F  /\  x  e.  dom  F )  -> 
( F `  x
)  e.  _V )
43funfni 5298 . . . . 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 2718 . 2  |-  ( F  Fn  A  ->  { x  e.  A  |  ( F `  x )  e.  ( _V  \  { B } ) }  =  { x  e.  A  |  ( F `  x )  =/=  B } )
81, 7eqtrd 2203 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 1348    e. wcel 2141    =/= wne 2340   {crab 2452   _Vcvv 2730    \ cdif 3118   {csn 3583   `'ccnv 4610   "cima 4614    Fn wfn 5193   ` cfv 5198
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-fv 5206
This theorem is referenced by: (None)
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