Description: The converse of an injective function is bijective. (Contributed by FL, 11-Nov-2011)
Ref | Expression | ||
---|---|---|---|
Assertion | f1cnv | ⊢ ( 𝐹 : 𝐴 –1-1→ 𝐵 → ◡ 𝐹 : ran 𝐹 –1-1-onto→ 𝐴 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1f1orn | ⊢ ( 𝐹 : 𝐴 –1-1→ 𝐵 → 𝐹 : 𝐴 –1-1-onto→ ran 𝐹 ) | |
2 | f1ocnv | ⊢ ( 𝐹 : 𝐴 –1-1-onto→ ran 𝐹 → ◡ 𝐹 : ran 𝐹 –1-1-onto→ 𝐴 ) | |
3 | 1 2 | syl | ⊢ ( 𝐹 : 𝐴 –1-1→ 𝐵 → ◡ 𝐹 : ran 𝐹 –1-1-onto→ 𝐴 ) |