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Definition df-fv 5641
Description: Define the value of a function, (𝐹𝐴), also known as function application. For example, (cos‘0) = 1 (we prove this in cos0 14364 after we define cosine in df-cos 14284). Typically, function 𝐹 is defined using maps-to notation (see df-mpt 4495 and df-mpt2 6368), but this is not required. For example, 𝐹 = {⟨2, 6⟩, ⟨3, 9⟩} → (𝐹‘3) = 9 (ex-fv 26053). Note that df-ov 6366 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 5952 and fvprc 5921). 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 6005, dffv3 5923, fv2 5922, and fv3 5940 (the latter two previously required 𝐴 to be a set.) Restricted equivalents that require 𝐹 to be a function are shown in funfv 5999 and funfv2 6000. For the familiar definition of function value in terms of ordered pair membership, see funopfvb 5973. (Contributed by NM, 1-Aug-1994.) Revised to use . Original version is now theorem dffv4 5924. (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 5633 . 2 class (𝐹𝐴)
4 vx . . . . 5 setvar 𝑥
54cv 1467 . . . 4 class 𝑥
61, 5, 2wbr 4434 . . 3 wff 𝐴𝐹𝑥
76, 4cio 5595 . 2 class (℩𝑥𝐴𝐹𝑥)
83, 7wceq 1468 1 wff (𝐹𝐴) = (℩𝑥𝐴𝐹𝑥)
Colors of variables: wff setvar class
This definition is referenced by:  tz6.12-2  5918  fveu  5919  fv2  5922  dffv3  5923  fveq1  5926  fveq2  5927  nffv  5934  fvex  5937  fvres  5941  tz6.12-1  5944  csbfv12  5965  fvopab5  6041  ovtpos  7065  rlimdm  13775  zsum  13944  isumclim3  13980  isumshft  14057  zprod  14151  iprodclim3  14213  avril1  26061  uncov  32159  fvsb  37162  dfafv2  39154  rlimdmafv  39199
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