Definition df-ac 10038

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 10381 as our definition, because the equivalence to more standard forms (dfac2 10054) 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 10381 itself as dfac0 10056. (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 10037	. 2 wff CHOICE
2		vf	. . . . . . 7 setvar 𝑓
3	2	cv 1541	. . . . . 6 class 𝑓
4		vx	. . . . . . 7 setvar 𝑥
5	4	cv 1541	. . . . . 6 class 𝑥
6	3, 5	wss 3903	. . . . 5 wff 𝑓 ⊆ 𝑥
7	5	cdm 5632	. . . . . 6 class dom 𝑥
8	3, 7	wfn 6495	. . . . 5 wff 𝑓 Fn dom 𝑥
9	6, 8	wa 395	. . . 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 1540	. 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  10043  ac7  10395  fineqvac  35294