Metamath Proof Explorer


Theorem f1ovi

Description: The identity relation is a one-to-one onto function on the universe. (Contributed by NM, 16-May-2004)

Ref Expression
Assertion f1ovi I : V –1-1-onto→ V

Proof

Step Hyp Ref Expression
1 f1oi ( I ↾ V ) : V –1-1-onto→ V
2 reli Rel I
3 dfrel3 ( Rel I ↔ ( I ↾ V ) = I )
4 2 3 mpbi ( I ↾ V ) = I
5 f1oeq1 ( ( I ↾ V ) = I → ( ( I ↾ V ) : V –1-1-onto→ V ↔ I : V –1-1-onto→ V ) )
6 4 5 ax-mp ( ( I ↾ V ) : V –1-1-onto→ V ↔ I : V –1-1-onto→ V )
7 1 6 mpbi I : V –1-1-onto→ V