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Definition df-fv 5858
Description: Define the value of a function, (𝐹𝐴), also known as function application. For example, (cos‘0) = 1 (we prove this in cos0 14800 after we define cosine in df-cos 14721). Typically, function 𝐹 is defined using maps-to notation (see df-mpt 4680 and df-mpt2 6610), but this is not required. For example, 𝐹 = {⟨2, 6⟩, ⟨3, 9⟩} → (𝐹‘3) = 9 (ex-fv 27148). Note that df-ov 6608 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 6176 and fvprc 6144). 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 6229, dffv3 6146, fv2 6145, and fv3 6164 (the latter two previously required 𝐴 to be a set.) Restricted equivalents that require 𝐹 to be a function are shown in funfv 6223 and funfv2 6224. For the familiar definition of function value in terms of ordered pair membership, see funopfvb 6197. (Contributed by NM, 1-Aug-1994.) Revised to use . Original version is now theorem dffv4 6147. (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 5850 . 2 class (𝐹𝐴)
4 vx . . . . 5 setvar 𝑥
54cv 1479 . . . 4 class 𝑥
61, 5, 2wbr 4618 . . 3 wff 𝐴𝐹𝑥
76, 4cio 5811 . 2 class (℩𝑥𝐴𝐹𝑥)
83, 7wceq 1480 1 wff (𝐹𝐴) = (℩𝑥𝐴𝐹𝑥)
Colors of variables: wff setvar class
This definition is referenced by:  tz6.12-2  6141  fveu  6142  fv2  6145  dffv3  6146  fveq1  6149  fveq2  6150  nffv  6157  fvex  6160  fvres  6165  tz6.12-1  6168  csbfv12  6189  fvopab5  6266  ovtpos  7313  rlimdm  14211  zsum  14377  isumclim3  14413  isumshft  14491  zprod  14587  iprodclim3  14651  avril1  27167  uncov  32989  fvsb  38105  dfafv2  40484  rlimdmafv  40529
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