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Definition df-fv 6557
Description: Define the value of a function, (𝐹𝐴), also known as function application. For example, (cos‘0) = 1 (we prove this in cos0 16130 after we define cosine in df-cos 16050). Typically, function 𝐹 is defined using maps-to notation (see df-mpt 5233 and df-mpo 7424), but this is not required. For example, 𝐹 = {⟨2, 6⟩, ⟨3, 9⟩} → (𝐹‘3) = 9 (ex-fv 30325). Note that df-ov 7422 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 6931 and fvprc 6888). 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 6992, dffv3 6892, fv2 6891, and fv3 6914 (the latter two previously required 𝐴 to be a set.) Restricted equivalents that require 𝐹 to be a function are shown in funfv 6984 and funfv2 6985. For the familiar definition of function value in terms of ordered pair membership, see funopfvb 6952. (Contributed by NM, 1-Aug-1994.) Revised to use . Original version is now Theorem dffv4 6893. (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 6549 . 2 class (𝐹𝐴)
4 vx . . . . 5 setvar 𝑥
54cv 1532 . . . 4 class 𝑥
61, 5, 2wbr 5149 . . 3 wff 𝐴𝐹𝑥
76, 4cio 6499 . 2 class (℩𝑥𝐴𝐹𝑥)
83, 7wceq 1533 1 wff (𝐹𝐴) = (℩𝑥𝐴𝐹𝑥)
Colors of variables: wff setvar class
This definition is referenced by:  tz6.12-2  6884  fveu  6885  fv2  6891  dffv3  6892  fveq1  6895  fveq2  6896  nffv  6906  fvex  6909  fvres  6915  tz6.12c  6918  tz6.12-1OLD  6920  csbfv12  6944  fvopab5  7037  ovtpos  8247  rlimdm  15531  zsum  15700  isumclim3  15741  isumshft  15821  zprod  15917  iprodclim3  15980  avril1  30345  uncov  37205  sn-tz6.12-2  42240  fvsb  44031  dfafv2  46650  rlimdmafv  46695  dfafv22  46777
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