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Definition df-fv 5884
Description: Define the value of a function, (𝐹𝐴), also known as function application. For example, (cos‘0) = 1 (we prove this in cos0 14861 after we define cosine in df-cos 14782). Typically, function 𝐹 is defined using maps-to notation (see df-mpt 4721 and df-mpt2 6640), but this is not required. For example, 𝐹 = {⟨2, 6⟩, ⟨3, 9⟩} → (𝐹‘3) = 9 (ex-fv 27270). Note that df-ov 6638 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 6205 and fvprc 6172). 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 6258, dffv3 6174, fv2 6173, and fv3 6193 (the latter two previously required 𝐴 to be a set.) Restricted equivalents that require 𝐹 to be a function are shown in funfv 6252 and funfv2 6253. For the familiar definition of function value in terms of ordered pair membership, see funopfvb 6226. (Contributed by NM, 1-Aug-1994.) Revised to use . Original version is now theorem dffv4 6175. (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 5876 . 2 class (𝐹𝐴)
4 vx . . . . 5 setvar 𝑥
54cv 1480 . . . 4 class 𝑥
61, 5, 2wbr 4644 . . 3 wff 𝐴𝐹𝑥
76, 4cio 5837 . 2 class (℩𝑥𝐴𝐹𝑥)
83, 7wceq 1481 1 wff (𝐹𝐴) = (℩𝑥𝐴𝐹𝑥)
Colors of variables: wff setvar class
This definition is referenced by:  tz6.12-2  6169  fveu  6170  fv2  6173  dffv3  6174  fveq1  6177  fveq2  6178  nffv  6185  fvex  6188  fvres  6194  tz6.12-1  6197  csbfv12  6218  fvopab5  6295  ovtpos  7352  rlimdm  14263  zsum  14430  isumclim3  14471  isumshft  14552  zprod  14648  iprodclim3  14712  avril1  27289  uncov  33361  fvsb  38476  dfafv2  40975  rlimdmafv  41020
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