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Definition df-fv 5263
Description: Define the value of a function,  ( F `  A ), also known as function application. For example,  ( cos `  0
)  =  1 (we prove this in cos0 12430 after we define cosine in df-cos 12352). Typically function  F is defined using maps-to notation (see df-mpt 4079 and df-mpt2 5863), but this is not required. For example,  F  =  { <. 2 ,  6 >. ,  <. 3 ,  9
>. }  ->  ( F `  3 )  =  9 (ex-fv 20830). Note that df-ov 5861 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 5552 and fvprc 5519). 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 5592, dffv3 5521, fv2 5520, and fv3 5541 (the latter two previously required  A to be a set.) Restricted equivalents that require  F to be a function are shown in funfv 5586 and funfv2 5587. For the familiar definition of function value in terms of ordered pair membership, see funopfvb 5566. (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 5255 . 2  class  ( F `
 A )
4 vx . . . . 5  set  x
54cv 1622 . . . 4  class  x
61, 5, 2wbr 4023 . . 3  wff  A F x
76, 4cio 5217 . 2  class  ( iota
x A F x )
83, 7wceq 1623 1  wff  ( F `
 A )  =  ( iota x A F x )
Colors of variables: wff set class
This definition is referenced by:  tz6.12-2  5516  fveu  5517  fv2  5520  dffv3  5521  fveq1  5524  fveq2  5525  nffv  5532  csbfv12g  5535  fvex  5539  fvres  5542  tz6.12-1  5544  ovtpos  6249  fvopab5  6289  rlimdm  12025  zsum  12191  isumclim3  12222  isumshft  12298  avril1  20836  fvsb  27655
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