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Definition df-fv 6570
Description: Define the value of a function, (𝐹𝐴), also known as function application. For example, (cos‘0) = 1 (we prove this in cos0 16182 after we define cosine in df-cos 16102). Typically, function 𝐹 is defined using maps-to notation (see df-mpt 5231 and df-mpo 7435), but this is not required. For example, 𝐹 = {⟨2, 6⟩, ⟨3, 9⟩} → (𝐹‘3) = 9 (ex-fv 30471). Note that df-ov 7433 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 6941 and fvprc 6898). 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 7003, dffv3 6902, fv2 6901, and fv3 6924 (the latter two previously required 𝐴 to be a set.) Restricted equivalents that require 𝐹 to be a function are shown in funfv 6995 and funfv2 6996. For the familiar definition of function value in terms of ordered pair membership, see funopfvb 6962. (Contributed by NM, 1-Aug-1994.) Revised to use . Original version is now Theorem dffv4 6903. (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 6562 . 2 class (𝐹𝐴)
4 vx . . . . 5 setvar 𝑥
54cv 1535 . . . 4 class 𝑥
61, 5, 2wbr 5147 . . 3 wff 𝐴𝐹𝑥
76, 4cio 6513 . 2 class (℩𝑥𝐴𝐹𝑥)
83, 7wceq 1536 1 wff (𝐹𝐴) = (℩𝑥𝐴𝐹𝑥)
Colors of variables: wff setvar class
This definition is referenced by:  tz6.12-2  6894  fveu  6895  fv2  6901  dffv3  6902  fveq1  6905  fveq2  6906  nffv  6916  fvex  6919  fvres  6925  tz6.12c  6928  tz6.12-1OLD  6930  csbfv12  6954  fvopab5  7048  ovtpos  8264  rlimdm  15583  zsum  15750  isumclim3  15791  isumshft  15871  zprod  15969  iprodclim3  16032  avril1  30491  uncov  37587  sn-tz6.12-2  42666  fvsb  44447  dfafv2  47081  rlimdmafv  47126  dfafv22  47208
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