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

Definition df-ac 9534
Description: The expression CHOICE will be used as a readable shorthand for any form of the axiom of choice; all concrete forms are long, cryptic, have dummy variables, or all three, making it useful to have a short name. Similar to the Axiom of Choice (first form) of [Enderton] p. 49.

There is a slight problem with taking the exact form of ax-ac 9873 as our definition, because the equivalence to more standard forms (dfac2 9549) requires the Axiom of Regularity, which we often try to avoid. Thus, we take the first of the "textbook forms" as the definition and derive the form of ax-ac 9873 itself as dfac0 9551. (Contributed by Mario Carneiro, 22-Feb-2015.)

Assertion
Ref Expression
df-ac (CHOICE ↔ ∀𝑥𝑓(𝑓𝑥𝑓 Fn dom 𝑥))
Distinct variable group:   𝑥,𝑓

Detailed syntax breakdown of Definition df-ac
StepHypRef Expression
1 wac 9533 . 2 wff CHOICE
2 vf . . . . . . 7 setvar 𝑓
32cv 1530 . . . . . 6 class 𝑓
4 vx . . . . . . 7 setvar 𝑥
54cv 1530 . . . . . 6 class 𝑥
63, 5wss 3934 . . . . 5 wff 𝑓𝑥
75cdm 5548 . . . . . 6 class dom 𝑥
83, 7wfn 6343 . . . . 5 wff 𝑓 Fn dom 𝑥
96, 8wa 398 . . . 4 wff (𝑓𝑥𝑓 Fn dom 𝑥)
109, 2wex 1774 . . 3 wff 𝑓(𝑓𝑥𝑓 Fn dom 𝑥)
1110, 4wal 1529 . 2 wff 𝑥𝑓(𝑓𝑥𝑓 Fn dom 𝑥)
121, 11wb 208 1 wff (CHOICE ↔ ∀𝑥𝑓(𝑓𝑥𝑓 Fn dom 𝑥))
Colors of variables: wff setvar class
This definition is referenced by:  dfac3  9539  ac7  9887
  Copyright terms: Public domain W3C validator