Definition df-ac 9527

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 9870 as our definition, because the equivalence to more standard forms (dfac2 9542) 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 9870 itself as dfac0 9544. (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 9526	. 2 wff CHOICE
2		vf	. . . . . . 7 setvar 𝑓
3	2	cv 1537	. . . . . 6 class 𝑓
4		vx	. . . . . . 7 setvar 𝑥
5	4	cv 1537	. . . . . 6 class 𝑥
6	3, 5	wss 3881	. . . . 5 wff 𝑓 ⊆ 𝑥
7	5	cdm 5519	. . . . . 6 class dom 𝑥
8	3, 7	wfn 6319	. . . . 5 wff 𝑓 Fn dom 𝑥
9	6, 8	wa 399	. . . 4 wff (𝑓 ⊆ 𝑥 ∧ 𝑓 Fn dom 𝑥)
10	9, 2	wex 1781	. . 3 wff ∃𝑓(𝑓 ⊆ 𝑥 ∧ 𝑓 Fn dom 𝑥)
11	10, 4	wal 1536	. 2 wff ∀𝑥∃𝑓(𝑓 ⊆ 𝑥 ∧ 𝑓 Fn dom 𝑥)
12	1, 11	wb 209	1 wff (CHOICE ↔ ∀𝑥∃𝑓(𝑓 ⊆ 𝑥 ∧ 𝑓 Fn dom 𝑥))

This definition is referenced by:  dfac3  9532  ac7  9884