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Definition df-fv 5697
Description: Define the value of a function, (𝐹𝐴), also known as function application. For example, (cos‘0) = 1 (we prove this in cos0 14586 after we define cosine in df-cos 14507). Typically, function 𝐹 is defined using maps-to notation (see df-mpt 4543 and df-mpt2 6430), but this is not required. For example, 𝐹 = {⟨2, 6⟩, ⟨3, 9⟩} → (𝐹‘3) = 9 (ex-fv 26431). Note that df-ov 6428 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 6011 and fvprc 5980). 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 6064, dffv3 5982, fv2 5981, and fv3 5999 (the latter two previously required 𝐴 to be a set.) Restricted equivalents that require 𝐹 to be a function are shown in funfv 6058 and funfv2 6059. For the familiar definition of function value in terms of ordered pair membership, see funopfvb 6032. (Contributed by NM, 1-Aug-1994.) Revised to use . Original version is now theorem dffv4 5983. (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 5689 . 2 class (𝐹𝐴)
4 vx . . . . 5 setvar 𝑥
54cv 1473 . . . 4 class 𝑥
61, 5, 2wbr 4481 . . 3 wff 𝐴𝐹𝑥
76, 4cio 5651 . 2 class (℩𝑥𝐴𝐹𝑥)
83, 7wceq 1474 1 wff (𝐹𝐴) = (℩𝑥𝐴𝐹𝑥)
Colors of variables: wff setvar class
This definition is referenced by:  tz6.12-2  5977  fveu  5978  fv2  5981  dffv3  5982  fveq1  5985  fveq2  5986  nffv  5993  fvex  5996  fvres  6000  tz6.12-1  6003  csbfv12  6024  fvopab5  6100  ovtpos  7128  rlimdm  13994  zsum  14163  isumclim3  14199  isumshft  14277  zprod  14373  iprodclim3  14437  avril1  26450  uncov  32435  fvsb  37559  dfafv2  39756  rlimdmafv  39801
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