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Definition df-fv 6143
Description: Define the value of a function, (𝐹𝐴), also known as function application. For example, (cos‘0) = 1 (we prove this in cos0 15282 after we define cosine in df-cos 15203). Typically, function 𝐹 is defined using maps-to notation (see df-mpt 4966 and df-mpt2 6927), but this is not required. For example, 𝐹 = {⟨2, 6⟩, ⟨3, 9⟩} → (𝐹‘3) = 9 (ex-fv 27889). Note that df-ov 6925 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 6476 and fvprc 6439). 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 6531, dffv3 6442, fv2 6441, and fv3 6464 (the latter two previously required 𝐴 to be a set.) Restricted equivalents that require 𝐹 to be a function are shown in funfv 6525 and funfv2 6526. For the familiar definition of function value in terms of ordered pair membership, see funopfvb 6498. (Contributed by NM, 1-Aug-1994.) Revised to use . Original version is now theorem dffv4 6443. (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 6135 . 2 class (𝐹𝐴)
4 vx . . . . 5 setvar 𝑥
54cv 1600 . . . 4 class 𝑥
61, 5, 2wbr 4886 . . 3 wff 𝐴𝐹𝑥
76, 4cio 6097 . 2 class (℩𝑥𝐴𝐹𝑥)
83, 7wceq 1601 1 wff (𝐹𝐴) = (℩𝑥𝐴𝐹𝑥)
Colors of variables: wff setvar class
This definition is referenced by:  tz6.12-2  6436  fveu  6437  fv2  6441  dffv3  6442  fveq1  6445  fveq2  6446  nffv  6456  fvex  6459  fvres  6465  tz6.12-1  6468  csbfv12  6490  fvopab5  6572  ovtpos  7649  rlimdm  14690  zsum  14856  isumclim3  14895  isumshft  14975  zprod  15070  iprodclim3  15133  avril1  27908  uncov  34009  fvsb  39602  dfafv2  42165  rlimdmafv  42210  dfafv22  42292
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