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Definition df-fv 5798
Description: Define the value of a function, (𝐹𝐴), also known as function application. For example, (cos‘0) = 1 (we prove this in cos0 14665 after we define cosine in df-cos 14586). Typically, function 𝐹 is defined using maps-to notation (see df-mpt 4639 and df-mpt2 6532), but this is not required. For example, 𝐹 = {⟨2, 6⟩, ⟨3, 9⟩} → (𝐹‘3) = 9 (ex-fv 26458). Note that df-ov 6530 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 6113 and fvprc 6082). 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 6166, dffv3 6084, fv2 6083, and fv3 6101 (the latter two previously required 𝐴 to be a set.) Restricted equivalents that require 𝐹 to be a function are shown in funfv 6160 and funfv2 6161. For the familiar definition of function value in terms of ordered pair membership, see funopfvb 6134. (Contributed by NM, 1-Aug-1994.) Revised to use . Original version is now theorem dffv4 6085. (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 5790 . 2 class (𝐹𝐴)
4 vx . . . . 5 setvar 𝑥
54cv 1473 . . . 4 class 𝑥
61, 5, 2wbr 4577 . . 3 wff 𝐴𝐹𝑥
76, 4cio 5752 . 2 class (℩𝑥𝐴𝐹𝑥)
83, 7wceq 1474 1 wff (𝐹𝐴) = (℩𝑥𝐴𝐹𝑥)
Colors of variables: wff setvar class
This definition is referenced by:  tz6.12-2  6079  fveu  6080  fv2  6083  dffv3  6084  fveq1  6087  fveq2  6088  nffv  6095  fvex  6098  fvres  6102  tz6.12-1  6105  csbfv12  6126  fvopab5  6202  ovtpos  7231  rlimdm  14076  zsum  14242  isumclim3  14278  isumshft  14356  zprod  14452  iprodclim3  14516  avril1  26477  uncov  32356  fvsb  37473  dfafv2  39659  rlimdmafv  39704
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