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 |
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 |