Theorem funcnv

Description: The converse of a class is a function iff the class is single-rooted, which means that for any y in the range of A there is at most one x such that x A y . Definition of single-rooted in Enderton p. 43. See funcnv2 for a simpler version. (Contributed by NM, 13-Aug-2004)

	Ref	Expression
Assertion	funcnv	|- ( Fun `' A <-> A. y e. ran A E* x x A y )

Proof

Step	Hyp	Ref	Expression
1		vex	|- x e. _V
2		vex	|- y e. _V
3	1 2	brelrn	|- ( x A y -> y e. ran A )
4	3	pm4.71ri	|- ( x A y <-> ( y e. ran A /\ x A y ) )
5	4	mobii	|- ( E* x x A y <-> E* x ( y e. ran A /\ x A y ) )
6		moanimv	|- ( E* x ( y e. ran A /\ x A y ) <-> ( y e. ran A -> E* x x A y ) )
7	5 6	bitri	|- ( E* x x A y <-> ( y e. ran A -> E* x x A y ) )
8	7	albii	|- ( A. y E* x x A y <-> A. y ( y e. ran A -> E* x x A y ) )
9		funcnv2	|- ( Fun `' A <-> A. y E* x x A y )
10		df-ral	|- ( A. y e. ran A E* x x A y <-> A. y ( y e. ran A -> E* x x A y ) )
11	8 9 10	3bitr4i	|- ( Fun `' A <-> A. y e. ran A E* x x A y )