Theorem ac5

Description: An Axiom of Choice equivalent: there exists a function f (called a choice function) with domain A that maps each nonempty member of the domain to an element of that member. Axiom AC of BellMachover p. 488. Note that the assertion that f be a function is not necessary; see ac4 . (Contributed by NM, 29-Aug-1999)

	Ref	Expression
Hypothesis	ac5.1	|- A e. _V
Assertion	ac5	|- E. f ( f Fn A /\ A. x e. A ( x =/= (/) -> ( f ` x ) e. x ) )

Proof

Step	Hyp	Ref	Expression
1		ac5.1	|- A e. _V
2		fneq2	|- ( y = A -> ( f Fn y <-> f Fn A ) )
3		raleq	|- ( y = A -> ( A. x e. y ( x =/= (/) -> ( f ` x ) e. x ) <-> A. x e. A ( x =/= (/) -> ( f ` x ) e. x ) ) )
4	2 3	anbi12d	|- ( y = A -> ( ( f Fn y /\ A. x e. y ( x =/= (/) -> ( f ` x ) e. x ) ) <-> ( f Fn A /\ A. x e. A ( x =/= (/) -> ( f ` x ) e. x ) ) ) )
5	4	exbidv	|- ( y = A -> ( E. f ( f Fn y /\ A. x e. y ( x =/= (/) -> ( f ` x ) e. x ) ) <-> E. f ( f Fn A /\ A. x e. A ( x =/= (/) -> ( f ` x ) e. x ) ) ) )
6		dfac4	|- ( CHOICE <-> A. y E. f ( f Fn y /\ A. x e. y ( x =/= (/) -> ( f ` x ) e. x ) ) )
7	6	axaci	|- E. f ( f Fn y /\ A. x e. y ( x =/= (/) -> ( f ` x ) e. x ) )
8	1 5 7	vtocl	|- E. f ( f Fn A /\ A. x e. A ( x =/= (/) -> ( f ` x ) e. x ) )