Theorem dff12

Description: Alternate definition of a one-to-one function. (Contributed by NM, 31-Dec-1996)

	Ref	Expression
Assertion	dff12	⊢ ( 𝐹 : 𝐴 –1-1→ 𝐵 ↔ ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ ∀ 𝑦 ∃* 𝑥 𝑥 𝐹 𝑦 ) )

Proof

Step	Hyp	Ref	Expression
1		df-f1	⊢ ( 𝐹 : 𝐴 –1-1→ 𝐵 ↔ ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ Fun ◡ 𝐹 ) )
2		funcnv2	⊢ ( Fun ◡ 𝐹 ↔ ∀ 𝑦 ∃* 𝑥 𝑥 𝐹 𝑦 )
3	2	anbi2i	⊢ ( ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ Fun ◡ 𝐹 ) ↔ ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ ∀ 𝑦 ∃* 𝑥 𝑥 𝐹 𝑦 ) )
4	1 3	bitri	⊢ ( 𝐹 : 𝐴 –1-1→ 𝐵 ↔ ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ ∀ 𝑦 ∃* 𝑥 𝑥 𝐹 𝑦 ) )