Definition df-ac 7953

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 8295 as our definition, because the equivalence to more standard forms (dfac2 7967) 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 8295 itself as dfac0 7969. (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 7952	. 2 wff CHOICE
2		vf	. . . . . . 7 set f
3	2	cv 1648	. . . . . 6 class f
4		vx	. . . . . . 7 set x
5	4	cv 1648	. . . . . 6 class x
6	3, 5	wss 3280	. . . . 5 wff f C_ x
7	5	cdm 4837	. . . . . 6 class dom x
8	3, 7	wfn 5408	. . . . 5 wff f Fn dom x
9	6, 8	wa 359	. . . 4 wff ( f C_ x /\ f Fn dom x )
10	9, 2	wex 1547	. . 3 wff E. f ( f C_ x /\ f Fn dom x )
11	10, 4	wal 1546	. 2 wff A. x E. f ( f C_ x /\ f Fn dom x )
12	1, 11	wb 177	1 wff (CHOICE <-> A. x E. f ( f C_ x /\ f Fn dom x ) )

This definition is referenced by:  dfac3  7958  ac7  8309