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Definition df-ac 8899
 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 9241 as our definition, because the equivalence to more standard forms (dfac2 8913) 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 9241 itself as dfac0 8915. (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 8898 . 2 wff CHOICE
2 vf . . . . . . 7 setvar 𝑓
32cv 1479 . . . . . 6 class 𝑓
4 vx . . . . . . 7 setvar 𝑥
54cv 1479 . . . . . 6 class 𝑥
63, 5wss 3560 . . . . 5 wff 𝑓𝑥
75cdm 5084 . . . . . 6 class dom 𝑥
83, 7wfn 5852 . . . . 5 wff 𝑓 Fn dom 𝑥
96, 8wa 384 . . . 4 wff (𝑓𝑥𝑓 Fn dom 𝑥)
109, 2wex 1701 . . 3 wff 𝑓(𝑓𝑥𝑓 Fn dom 𝑥)
1110, 4wal 1478 . 2 wff 𝑥𝑓(𝑓𝑥𝑓 Fn dom 𝑥)
121, 11wb 196 1 wff (CHOICE ↔ ∀𝑥𝑓(𝑓𝑥𝑓 Fn dom 𝑥))
 Colors of variables: wff setvar class This definition is referenced by:  dfac3  8904  ac7  9255
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