Definition df-ac 8291

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 8633 as our definition, because the equivalence to more standard forms (dfac2 8305) 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 8633 itself as dfac0 8307. (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 8290	. 2 wff CHOICE
2		vf	. . . . . . 7 setvar f
3	2	cv 1368	. . . . . 6 class f
4		vx	. . . . . . 7 setvar x
5	4	cv 1368	. . . . . 6 class x
6	3, 5	wss 3333	. . . . 5 wff f C_ x
7	5	cdm 4845	. . . . . 6 class dom x
8	3, 7	wfn 5418	. . . . 5 wff f Fn dom x
9	6, 8	wa 369	. . . 4 wff ( f C_ x /\ f Fn dom x )
10	9, 2	wex 1586	. . 3 wff E. f ( f C_ x /\ f Fn dom x )
11	10, 4	wal 1367	. 2 wff A. x E. f ( f C_ x /\ f Fn dom x )
12	1, 11	wb 184	1 wff (CHOICE <-> A. x E. f ( f C_ x /\ f Fn dom x ) )

This definition is referenced by:  dfac3  8296  ac7  8647