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Definition df-fv 4937
Description: Define the value of a function,  ( F `  A ), also known as function application. For example,  (  _I  `  (/) )  =  (/). Typically, function  F is defined using maps-to notation (see df-mpt 3847), but this is not required. For example, F = {  <. 2 , 6  >.,  <. 3 , 9  >. } -> ( F  ` 3 ) = 9 . We will later 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. 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. (Contributed by NM, 1-Aug-1994.) Revised to use  iota. (Revised 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 4929 . 2  class  ( F `
 A )
4 vx . . . . 5  setvar  x
54cv 1258 . . . 4  class  x
61, 5, 2wbr 3791 . . 3  wff  A F x
76, 4cio 4892 . 2  class  ( iota
x A F x )
83, 7wceq 1259 1  wff  ( F `
 A )  =  ( iota x A F x )
Colors of variables: wff set class
This definition is referenced by:  tz6.12-2  5196  fveu  5197  fv2  5200  dffv3g  5201  fveq1  5204  fveq2  5205  nffv  5212  fvss  5216  funfvex  5219  fvres  5225  tz6.12-1  5227  nfvres  5233  0fv  5235  csbfv12g  5236  ovtposg  5904
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