Definition df-ac 10026

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 10369 as our definition, because the equivalence to more standard forms (dfac2 10042) 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 10369 itself as dfac0 10044. (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 10025	. 2 wff CHOICE
2		vf	. . . . . . 7 setvar 𝑓
3	2	cv 1540	. . . . . 6 class 𝑓
4		vx	. . . . . . 7 setvar 𝑥
5	4	cv 1540	. . . . . 6 class 𝑥
6	3, 5	wss 3901	. . . . 5 wff 𝑓 ⊆ 𝑥
7	5	cdm 5624	. . . . . 6 class dom 𝑥
8	3, 7	wfn 6487	. . . . 5 wff 𝑓 Fn dom 𝑥
9	6, 8	wa 395	. . . 4 wff (𝑓 ⊆ 𝑥 ∧ 𝑓 Fn dom 𝑥)
10	9, 2	wex 1780	. . 3 wff ∃𝑓(𝑓 ⊆ 𝑥 ∧ 𝑓 Fn dom 𝑥)
11	10, 4	wal 1539	. 2 wff ∀𝑥∃𝑓(𝑓 ⊆ 𝑥 ∧ 𝑓 Fn dom 𝑥)
12	1, 11	wb 206	1 wff (CHOICE ↔ ∀𝑥∃𝑓(𝑓 ⊆ 𝑥 ∧ 𝑓 Fn dom 𝑥))

This definition is referenced by:  dfac3  10031  ac7  10383  fineqvac  35272