Definition df-ac 8545

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 8887 as our definition, because the equivalence to more standard forms (dfac2 8559) 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 8887 itself as dfac0 8561. (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 8544	. 2 wff CHOICE
2		vf	. . . . . . 7 setvar f
3	2	cv 1436	. . . . . 6 class f
4		vx	. . . . . . 7 setvar x
5	4	cv 1436	. . . . . 6 class x
6	3, 5	wss 3442	. . . . 5 wff f C_ x
7	5	cdm 4854	. . . . . 6 class dom x
8	3, 7	wfn 5596	. . . . 5 wff f Fn dom x
9	6, 8	wa 370	. . . 4 wff ( f C_ x /\ f Fn dom x )
10	9, 2	wex 1659	. . 3 wff E. f ( f C_ x /\ f Fn dom x )
11	10, 4	wal 1435	. 2 wff A. x E. f ( f C_ x /\ f Fn dom x )
12	1, 11	wb 187	1 wff (CHOICE <-> A. x E. f ( f C_ x /\ f Fn dom x ) )

This definition is referenced by:  dfac3  8550  ac7  8901