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Definition df-fv 5402
Description: Define the value of a function,  ( F `  A ), also known as function application. For example,  ( cos `  0
)  =  1 (we prove this in cos0 12678 after we define cosine in df-cos 12600). Typically, function  F is defined using maps-to notation (see df-mpt 4209 and df-mpt2 6025), but this is not required. For example,  F  =  { <. 2 ,  6 >. ,  <. 3 ,  9
>. }  ->  ( F `  3 )  =  9 (ex-fv 21599). Note that df-ov 6023 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 5695 and fvprc 5662). 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 5735, dffv3 5664, fv2 5663, and fv3 5684 (the latter two previously required  A to be a set.) Restricted equivalents that require  F to be a function are shown in funfv 5729 and funfv2 5730. For the familiar definition of function value in terms of ordered pair membership, see funopfvb 5709. (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 5394 . 2  class  ( F `
 A )
4 vx . . . . 5  set  x
54cv 1648 . . . 4  class  x
61, 5, 2wbr 4153 . . 3  wff  A F x
76, 4cio 5356 . 2  class  ( iota
x A F x )
83, 7wceq 1649 1  wff  ( F `
 A )  =  ( iota x A F x )
Colors of variables: wff set class
This definition is referenced by:  tz6.12-2  5659  fveu  5660  fv2  5663  dffv3  5664  fveq1  5667  fveq2  5668  nffv  5675  csbfv12g  5678  fvex  5682  fvres  5685  tz6.12-1  5687  ovtpos  6430  fvopab5  6470  rlimdm  12272  zsum  12439  isumclim3  12470  isumshft  12546  avril1  21605  zprod  25042  iprodclim3  25085  fvsb  27323  dfafv2  27665  rlimdmafv  27710
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