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Definition df-fv 6542
Description: Define the value of a function, (𝐹𝐴), also known as function application. For example, (cos‘0) = 1 (we prove this in cos0 16096 after we define cosine in df-cos 16016). Typically, function 𝐹 is defined using maps-to notation (see df-mpt 5223 and df-mpo 7407), but this is not required. For example, 𝐹 = {⟨2, 6⟩, ⟨3, 9⟩} → (𝐹‘3) = 9 (ex-fv 30191). Note that df-ov 7405 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 6917 and fvprc 6874). 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 6977, dffv3 6878, fv2 6877, and fv3 6900 (the latter two previously required 𝐴 to be a set.) Restricted equivalents that require 𝐹 to be a function are shown in funfv 6969 and funfv2 6970. For the familiar definition of function value in terms of ordered pair membership, see funopfvb 6938. (Contributed by NM, 1-Aug-1994.) Revised to use . Original version is now Theorem dffv4 6879. (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 6534 . 2 class (𝐹𝐴)
4 vx . . . . 5 setvar 𝑥
54cv 1532 . . . 4 class 𝑥
61, 5, 2wbr 5139 . . 3 wff 𝐴𝐹𝑥
76, 4cio 6484 . 2 class (℩𝑥𝐴𝐹𝑥)
83, 7wceq 1533 1 wff (𝐹𝐴) = (℩𝑥𝐴𝐹𝑥)
Colors of variables: wff setvar class
This definition is referenced by:  tz6.12-2  6870  fveu  6871  fv2  6877  dffv3  6878  fveq1  6881  fveq2  6882  nffv  6892  fvex  6895  fvres  6901  tz6.12c  6904  tz6.12-1OLD  6906  csbfv12  6930  fvopab5  7021  ovtpos  8222  rlimdm  15497  zsum  15666  isumclim3  15707  isumshft  15787  zprod  15883  iprodclim3  15946  avril1  30211  uncov  36973  fvsb  43761  dfafv2  46386  rlimdmafv  46431  dfafv22  46513
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