Definition df-ac 8495

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 8837 as our definition, because the equivalence to more standard forms (dfac2 8509) 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 8837 itself as dfac0 8511. (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 8494	. 2 wff CHOICE
2		vf	. . . . . . 7 setvar f
3	2	cv 1380	. . . . . 6 class f
4		vx	. . . . . . 7 setvar x
5	4	cv 1380	. . . . . 6 class x
6	3, 5	wss 3458	. . . . 5 wff f C_ x
7	5	cdm 4985	. . . . . 6 class dom x
8	3, 7	wfn 5569	. . . . 5 wff f Fn dom x
9	6, 8	wa 369	. . . 4 wff ( f C_ x /\ f Fn dom x )
10	9, 2	wex 1597	. . 3 wff E. f ( f C_ x /\ f Fn dom x )
11	10, 4	wal 1379	. 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  8500  ac7  8851