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Definition df-fv 6556
Description: Define the value of a function, (𝐹𝐴), also known as function application. For example, (cos‘0) = 1 (we prove this in cos0 16127 after we define cosine in df-cos 16047). Typically, function 𝐹 is defined using maps-to notation (see df-mpt 5232 and df-mpo 7425), but this is not required. For example, 𝐹 = {⟨2, 6⟩, ⟨3, 9⟩} → (𝐹‘3) = 9 (ex-fv 30266). Note that df-ov 7423 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 6932 and fvprc 6889). 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 6993, dffv3 6893, fv2 6892, and fv3 6915 (the latter two previously required 𝐴 to be a set.) Restricted equivalents that require 𝐹 to be a function are shown in funfv 6985 and funfv2 6986. For the familiar definition of function value in terms of ordered pair membership, see funopfvb 6953. (Contributed by NM, 1-Aug-1994.) Revised to use . Original version is now Theorem dffv4 6894. (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 6548 . 2 class (𝐹𝐴)
4 vx . . . . 5 setvar 𝑥
54cv 1533 . . . 4 class 𝑥
61, 5, 2wbr 5148 . . 3 wff 𝐴𝐹𝑥
76, 4cio 6498 . 2 class (℩𝑥𝐴𝐹𝑥)
83, 7wceq 1534 1 wff (𝐹𝐴) = (℩𝑥𝐴𝐹𝑥)
Colors of variables: wff setvar class
This definition is referenced by:  tz6.12-2  6885  fveu  6886  fv2  6892  dffv3  6893  fveq1  6896  fveq2  6897  nffv  6907  fvex  6910  fvres  6916  tz6.12c  6919  tz6.12-1OLD  6921  csbfv12  6945  fvopab5  7038  ovtpos  8247  rlimdm  15528  zsum  15697  isumclim3  15738  isumshft  15818  zprod  15914  iprodclim3  15977  avril1  30286  uncov  37074  fvsb  43889  dfafv2  46512  rlimdmafv  46557  dfafv22  46639
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