This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

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 )