Definition df-ac 9252

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 9596 as our definition, because the equivalence to more standard forms (dfac2 9267) 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 9596 itself as dfac0 9270. (Contributed by Mario Carneiro, 22-Feb-2015.)

Assertion

Ref	Expression
df-ac	⊢ (CHOICE ↔ ∀𝑥∃𝑓(𝑓 ⊆ 𝑥 ∧ 𝑓 Fn dom 𝑥))

Distinct variable group:   𝑥,𝑓

Detailed syntax breakdown of Definition df-ac

Step	Hyp	Ref	Expression
1		wac 9251	. 2 wff CHOICE
2		vf	. . . . . . 7 setvar 𝑓
3	2	cv 1655	. . . . . 6 class 𝑓
4		vx	. . . . . . 7 setvar 𝑥
5	4	cv 1655	. . . . . 6 class 𝑥
6	3, 5	wss 3798	. . . . 5 wff 𝑓 ⊆ 𝑥
7	5	cdm 5342	. . . . . 6 class dom 𝑥
8	3, 7	wfn 6118	. . . . 5 wff 𝑓 Fn dom 𝑥
9	6, 8	wa 386	. . . 4 wff (𝑓 ⊆ 𝑥 ∧ 𝑓 Fn dom 𝑥)
10	9, 2	wex 1878	. . 3 wff ∃𝑓(𝑓 ⊆ 𝑥 ∧ 𝑓 Fn dom 𝑥)
11	10, 4	wal 1654	. 2 wff ∀𝑥∃𝑓(𝑓 ⊆ 𝑥 ∧ 𝑓 Fn dom 𝑥)
12	1, 11	wb 198	1 wff (CHOICE ↔ ∀𝑥∃𝑓(𝑓 ⊆ 𝑥 ∧ 𝑓 Fn dom 𝑥))

This definition is referenced by:  dfac3  9257  ac7  9610