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Definition df-fv 6039
Description: Define the value of a function, (𝐹𝐴), also known as function application. For example, (cos‘0) = 1 (we prove this in cos0 15086 after we define cosine in df-cos 15007). Typically, function 𝐹 is defined using maps-to notation (see df-mpt 4864 and df-mpt2 6798), but this is not required. For example, 𝐹 = {⟨2, 6⟩, ⟨3, 9⟩} → (𝐹‘3) = 9 (ex-fv 27642). Note that df-ov 6796 will define two-argument functions using ordered pairs as (𝐴𝐹𝐵) = (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 6359 and fvprc 6326). 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 6413, dffv3 6328, fv2 6327, and fv3 6347 (the latter two previously required 𝐴 to be a set.) Restricted equivalents that require 𝐹 to be a function are shown in funfv 6407 and funfv2 6408. For the familiar definition of function value in terms of ordered pair membership, see funopfvb 6380. (Contributed by NM, 1-Aug-1994.) Revised to use . Original version is now theorem dffv4 6329. (Revised by Scott Fenton, 6-Oct-2017.)
Assertion
Ref Expression
df-fv (𝐹𝐴) = (℩𝑥𝐴𝐹𝑥)
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 6031 . 2 class (𝐹𝐴)
4 vx . . . . 5 setvar 𝑥
54cv 1630 . . . 4 class 𝑥
61, 5, 2wbr 4786 . . 3 wff 𝐴𝐹𝑥
76, 4cio 5992 . 2 class (℩𝑥𝐴𝐹𝑥)
83, 7wceq 1631 1 wff (𝐹𝐴) = (℩𝑥𝐴𝐹𝑥)
Colors of variables: wff setvar class
This definition is referenced by:  tz6.12-2  6323  fveu  6324  fv2  6327  dffv3  6328  fveq1  6331  fveq2  6332  nffv  6339  fvex  6342  fvres  6348  tz6.12-1  6351  csbfv12  6372  fvopab5  6452  ovtpos  7519  rlimdm  14490  zsum  14657  isumclim3  14698  isumshft  14778  zprod  14874  iprodclim3  14937  avril1  27661  uncov  33723  fvsb  39181  dfafv2  41732  rlimdmafv  41777
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