Definition df-ac 8493

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 8835 as our definition, because the equivalence to more standard forms (dfac2 8507) 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 8835 itself as dfac0 8509. (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 8492	. 2 wff CHOICE
2		vf	. . . . . . 7 setvar f
3	2	cv 1378	. . . . . 6 class f
4		vx	. . . . . . 7 setvar x
5	4	cv 1378	. . . . . 6 class x
6	3, 5	wss 3476	. . . . 5 wff f C_ x
7	5	cdm 4999	. . . . . 6 class dom x
8	3, 7	wfn 5581	. . . . 5 wff f Fn dom x
9	6, 8	wa 369	. . . 4 wff ( f C_ x /\ f Fn dom x )
10	9, 2	wex 1596	. . 3 wff E. f ( f C_ x /\ f Fn dom x )
11	10, 4	wal 1377	. 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  8498  ac7  8849