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Definition df-fv 5265
Description: Define the value of a function,  ( F `  A ), also known as function application. For example,  ( cos `  0
)  =  1 (we prove this in cos0 12432 after we define cosine in df-cos 12354). Typically, function  F is defined using maps-to notation (see df-mpt 4081 and df-mpt2 5865), but this is not required. For example,  F  =  { <. 2 ,  6 >. ,  <. 3 ,  9
>. }  ->  ( F `  3 )  =  9 (ex-fv 20832). Note that df-ov 5863 will define two-argument functions using ordered pairs as  ( A F B )  =  ( F `  <. A ,  B >. ). This particular definition is quite convenient: it can be applied to any class and evaluates to the empty set when it is not meaningful (as shown by ndmfv 5554 and fvprc 5521). The left apostrophe notation originated with Peano and was adopted in Definition *30.01 of [WhiteheadRussell] p. 235, Definition 10.11 of [Quine] p. 68, and Definition 6.11 of [TakeutiZaring] p. 26. It means the same thing as the more familiar  F ( A ) notation for a function's value at  A, i.e. " F of  A," but without context-dependent notational ambiguity. Alternate definitions are dffv2 5594, dffv3 5523, fv2 5522, and fv3 5543 (the latter two previously required  A to be a set.) Restricted equivalents that require  F to be a function are shown in funfv 5588 and funfv2 5589. For the familiar definition of function value in terms of ordered pair membership, see funopfvb 5568. (Contributed by Scott Fenton, 6-Oct-2017.)
Assertion
Ref Expression
df-fv  |-  ( F `
 A )  =  ( iota x A F x )
Distinct variable groups:    x, A    x, F

Detailed syntax breakdown of Definition df-fv
StepHypRef Expression
1 cA . . 3  class  A
2 cF . . 3  class  F
31, 2cfv 5257 . 2  class  ( F `
 A )
4 vx . . . . 5  set  x
54cv 1624 . . . 4  class  x
61, 5, 2wbr 4025 . . 3  wff  A F x
76, 4cio 5219 . 2  class  ( iota
x A F x )
83, 7wceq 1625 1  wff  ( F `
 A )  =  ( iota x A F x )
Colors of variables: wff set class
This definition is referenced by:  tz6.12-2  5518  fveu  5519  fv2  5522  dffv3  5523  fveq1  5526  fveq2  5527  nffv  5534  csbfv12g  5537  fvex  5541  fvres  5544  tz6.12-1  5546  ovtpos  6251  fvopab5  6291  rlimdm  12027  zsum  12193  isumclim3  12224  isumshft  12300  avril1  20838  fvsb  27666
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