Definition df-ac 8939

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 9281 as our definition, because the equivalence to more standard forms (dfac2 8953) 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 9281 itself as dfac0 8955. (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 8938	. 2 wff CHOICE
2		vf	. . . . . . 7 setvar 𝑓
3	2	cv 1482	. . . . . 6 class 𝑓
4		vx	. . . . . . 7 setvar 𝑥
5	4	cv 1482	. . . . . 6 class 𝑥
6	3, 5	wss 3574	. . . . 5 wff 𝑓 ⊆ 𝑥
7	5	cdm 5114	. . . . . 6 class dom 𝑥
8	3, 7	wfn 5883	. . . . 5 wff 𝑓 Fn dom 𝑥
9	6, 8	wa 384	. . . 4 wff (𝑓 ⊆ 𝑥 ∧ 𝑓 Fn dom 𝑥)
10	9, 2	wex 1704	. . 3 wff ∃𝑓(𝑓 ⊆ 𝑥 ∧ 𝑓 Fn dom 𝑥)
11	10, 4	wal 1481	. 2 wff ∀𝑥∃𝑓(𝑓 ⊆ 𝑥 ∧ 𝑓 Fn dom 𝑥)
12	1, 11	wb 196	1 wff (CHOICE ↔ ∀𝑥∃𝑓(𝑓 ⊆ 𝑥 ∧ 𝑓 Fn dom 𝑥))

This definition is referenced by:  dfac3  8944  ac7  9295