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Definition df-fv 5934
Description: Define the value of a function, (𝐹𝐴), also known as function application. For example, (cos‘0) = 1 (we prove this in cos0 14924 after we define cosine in df-cos 14845). Typically, function 𝐹 is defined using maps-to notation (see df-mpt 4763 and df-mpt2 6695), but this is not required. For example, 𝐹 = {⟨2, 6⟩, ⟨3, 9⟩} → (𝐹‘3) = 9 (ex-fv 27430). Note that df-ov 6693 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 6256 and fvprc 6223). 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 6310, dffv3 6225, fv2 6224, and fv3 6244 (the latter two previously required 𝐴 to be a set.) Restricted equivalents that require 𝐹 to be a function are shown in funfv 6304 and funfv2 6305. For the familiar definition of function value in terms of ordered pair membership, see funopfvb 6277. (Contributed by NM, 1-Aug-1994.) Revised to use . Original version is now theorem dffv4 6226. (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 5926 . 2 class (𝐹𝐴)
4 vx . . . . 5 setvar 𝑥
54cv 1522 . . . 4 class 𝑥
61, 5, 2wbr 4685 . . 3 wff 𝐴𝐹𝑥
76, 4cio 5887 . 2 class (℩𝑥𝐴𝐹𝑥)
83, 7wceq 1523 1 wff (𝐹𝐴) = (℩𝑥𝐴𝐹𝑥)
Colors of variables: wff setvar class
This definition is referenced by:  tz6.12-2  6220  fveu  6221  fv2  6224  dffv3  6225  fveq1  6228  fveq2  6229  nffv  6236  fvex  6239  fvres  6245  tz6.12-1  6248  csbfv12  6269  fvopab5  6349  ovtpos  7412  rlimdm  14326  zsum  14493  isumclim3  14534  isumshft  14615  zprod  14711  iprodclim3  14775  avril1  27449  uncov  33520  fvsb  38973  dfafv2  41533  rlimdmafv  41578
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