Theorem f1oen

Description: The domain and range of a one-to-one, onto function are equinumerous. (Contributed by NM, 19-Jun-1998)

	Ref	Expression
Hypothesis	f1oen.1	⊢ 𝐴 ∈ V
Assertion	f1oen	⊢ ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 → 𝐴 ≈ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		f1oen.1	⊢ 𝐴 ∈ V
2		f1oeng	⊢ ( ( 𝐴 ∈ V ∧ 𝐹 : 𝐴 –1-1-onto→ 𝐵 ) → 𝐴 ≈ 𝐵 )
3	1 2	mpan	⊢ ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 → 𝐴 ≈ 𝐵 )