Definition df-ac 8573

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 8915 as our definition, because the equivalence to more standard forms (dfac2 8587) 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 8915 itself as dfac0 8589. (Contributed by Mario Carneiro, 22-Feb-2015.)

Assertion

Ref	Expression
df-ac	|- (CHOICE <-> A. x E. f ( f C_ x /\ f Fn dom x ) )

Distinct variable group:   x, f

Detailed syntax breakdown of Definition df-ac

Step	Hyp	Ref	Expression
1		wac 8572	. 2 wff CHOICE
2		vf	. . . . . . 7 setvar f
3	2	cv 1454	. . . . . 6 class f
4		vx	. . . . . . 7 setvar x
5	4	cv 1454	. . . . . 6 class x
6	3, 5	wss 3416	. . . . 5 wff f C_ x
7	5	cdm 4853	. . . . . 6 class dom x
8	3, 7	wfn 5596	. . . . 5 wff f Fn dom x
9	6, 8	wa 375	. . . 4 wff ( f C_ x /\ f Fn dom x )
10	9, 2	wex 1674	. . 3 wff E. f ( f C_ x /\ f Fn dom x )
11	10, 4	wal 1453	. 2 wff A. x E. f ( f C_ x /\ f Fn dom x )
12	1, 11	wb 189	1 wff (CHOICE <-> A. x E. f ( f C_ x /\ f Fn dom x ) )

This definition is referenced by:  dfac3  8578  ac7  8929