Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  df-fv Structured version   Visualization version   GIF version

Definition df-fv 6356
 Description: Define the value of a function, (𝐹‘𝐴), also known as function application. For example, (cos‘0) = 1 (we prove this in cos0 15495 after we define cosine in df-cos 15416). Typically, function 𝐹 is defined using maps-to notation (see df-mpt 5138 and df-mpo 7153), but this is not required. For example, 𝐹 = {⟨2, 6⟩, ⟨3, 9⟩} → (𝐹‘3) = 9 (ex-fv 28214). Note that df-ov 7151 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 6693 and fvprc 6656). 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 6749, dffv3 6659, fv2 6658, and fv3 6681 (the latter two previously required 𝐴 to be a set.) Restricted equivalents that require 𝐹 to be a function are shown in funfv 6743 and funfv2 6744. For the familiar definition of function value in terms of ordered pair membership, see funopfvb 6714. (Contributed by NM, 1-Aug-1994.) Revised to use ℩. Original version is now theorem dffv4 6660. (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 6348 . 2 class (𝐹𝐴)
4 vx . . . . 5 setvar 𝑥
54cv 1529 . . . 4 class 𝑥
61, 5, 2wbr 5057 . . 3 wff 𝐴𝐹𝑥
76, 4cio 6305 . 2 class (℩𝑥𝐴𝐹𝑥)
83, 7wceq 1530 1 wff (𝐹𝐴) = (℩𝑥𝐴𝐹𝑥)
 Colors of variables: wff setvar class This definition is referenced by:  tz6.12-2  6653  fveu  6654  fv2  6658  dffv3  6659  fveq1  6662  fveq2  6663  nffv  6673  fvex  6676  fvres  6682  tz6.12-1  6685  csbfv12  6706  fvopab5  6793  ovtpos  7899  rlimdm  14900  zsum  15067  isumclim3  15106  isumshft  15186  zprod  15283  iprodclim3  15346  avril1  28234  uncov  34865  fnimasnd  39111  fvsb  40774  dfafv2  43321  rlimdmafv  43366  dfafv22  43448
 Copyright terms: Public domain W3C validator