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Definition df-fv 6357
Description: Define the value of a function, (𝐹𝐴), also known as function application. For example, (cos‘0) = 1 (we prove this in cos0 15493 after we define cosine in df-cos 15414). Typically, function 𝐹 is defined using maps-to notation (see df-mpt 5139 and df-mpo 7150), but this is not required. For example, 𝐹 = {⟨2, 6⟩, ⟨3, 9⟩} → (𝐹‘3) = 9 (ex-fv 28150). Note that df-ov 7148 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 6694 and fvprc 6657). 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 6750, dffv3 6660, fv2 6659, and fv3 6682 (the latter two previously required 𝐴 to be a set.) Restricted equivalents that require 𝐹 to be a function are shown in funfv 6744 and funfv2 6745. For the familiar definition of function value in terms of ordered pair membership, see funopfvb 6715. (Contributed by NM, 1-Aug-1994.) Revised to use . Original version is now theorem dffv4 6661. (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 6349 . 2 class (𝐹𝐴)
4 vx . . . . 5 setvar 𝑥
54cv 1527 . . . 4 class 𝑥
61, 5, 2wbr 5058 . . 3 wff 𝐴𝐹𝑥
76, 4cio 6306 . 2 class (℩𝑥𝐴𝐹𝑥)
83, 7wceq 1528 1 wff (𝐹𝐴) = (℩𝑥𝐴𝐹𝑥)
Colors of variables: wff setvar class
This definition is referenced by:  tz6.12-2  6654  fveu  6655  fv2  6659  dffv3  6660  fveq1  6663  fveq2  6664  nffv  6674  fvex  6677  fvres  6683  tz6.12-1  6686  csbfv12  6707  fvopab5  6793  ovtpos  7898  rlimdm  14898  zsum  15065  isumclim3  15104  isumshft  15184  zprod  15281  iprodclim3  15344  avril1  28170  uncov  34755  fnimasnd  39001  fvsb  40664  dfafv2  43212  rlimdmafv  43257  dfafv22  43339
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