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Definition df-fv 4608
Description: Define the value of a function,  ( F `  A ), also known as function application. For example,  ( cos `  0
)  =  1 (we prove this in cos0 12304 after we define cosine in df-cos 12226). Typically function  F is defined using maps-to notation (see df-mpt 3976 and df-mpt2 5715), but this is not required. For example,  F  =  { <. 2 ,  6 >. ,  <. 3 ,  9
>. }  ->  ( F `  3 )  =  9 (ex-fv 20643). Note that df-ov 5713 will define two-argument functions using ordered pairs as  ( A F B )  =  ( F `  <. A ,  B >. ). Although based on the idea embodied by Definition 10.2 of [Quine] p. 65 (see args 4948), our definition apparently does not appear in the literature. However, it 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 5405 and fvprc 5374). 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 5444 and dffv3 6171. For other alternate definitions (that fail to evaluate to the empty set for proper classes), see fv2 5373, fv3 5393, and fv4 6172. Restricted equivalents that require  F to be a function are shown in funfv 5438 and funfv2 5439. For the familiar definition of function value in terms of ordered pair membership, see funopfvb 5418. (Contributed by NM, 1-Aug-1994.)
Assertion
Ref Expression
df-fv  |-  ( F `
 A )  = 
U. { x  |  ( F " { A } )  =  {
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 4592 . 2  class  ( F `
 A )
41csn 3544 . . . . . 6  class  { A }
52, 4cima 4583 . . . . 5  class  ( F
" { A }
)
6 vx . . . . . . 7  set  x
76cv 1618 . . . . . 6  class  x
87csn 3544 . . . . 5  class  { x }
95, 8wceq 1619 . . . 4  wff  ( F
" { A }
)  =  { x }
109, 6cab 2239 . . 3  class  { x  |  ( F " { A } )  =  { x } }
1110cuni 3727 . 2  class  U. {
x  |  ( F
" { A }
)  =  { x } }
123, 11wceq 1619 1  wff  ( F `
 A )  = 
U. { x  |  ( F " { A } )  =  {
x } }
Colors of variables: wff set class
This definition is referenced by:  fv2  5373  fvprc  5374  fveq1  5376  fveq2  5377  nffv  5384  csbfv12g  5387  csbfv12gALT  5388  fvex  5391  fvres  5394  fvco2  5446  dffv3  6171  shftval  11446  avril1  20666  repfuntw  24326  csbfv12gALTVD  27365
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