Theorem f1ss

Description: A function that is one-to-one is also one-to-one on some superset of its codomain. (Contributed by Mario Carneiro, 12-Jan-2013)

	Ref	Expression
Assertion	f1ss	⊢ ( ( 𝐹 : 𝐴 –1-1→ 𝐵 ∧ 𝐵 ⊆ 𝐶 ) → 𝐹 : 𝐴 –1-1→ 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		f1f	⊢ ( 𝐹 : 𝐴 –1-1→ 𝐵 → 𝐹 : 𝐴 ⟶ 𝐵 )
2		fss	⊢ ( ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ 𝐵 ⊆ 𝐶 ) → 𝐹 : 𝐴 ⟶ 𝐶 )
3	1 2	sylan	⊢ ( ( 𝐹 : 𝐴 –1-1→ 𝐵 ∧ 𝐵 ⊆ 𝐶 ) → 𝐹 : 𝐴 ⟶ 𝐶 )
4		df-f1	⊢ ( 𝐹 : 𝐴 –1-1→ 𝐵 ↔ ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ Fun ◡ 𝐹 ) )
5	4	simprbi	⊢ ( 𝐹 : 𝐴 –1-1→ 𝐵 → Fun ◡ 𝐹 )
6	5	adantr	⊢ ( ( 𝐹 : 𝐴 –1-1→ 𝐵 ∧ 𝐵 ⊆ 𝐶 ) → Fun ◡ 𝐹 )
7		df-f1	⊢ ( 𝐹 : 𝐴 –1-1→ 𝐶 ↔ ( 𝐹 : 𝐴 ⟶ 𝐶 ∧ Fun ◡ 𝐹 ) )
8	3 6 7	sylanbrc	⊢ ( ( 𝐹 : 𝐴 –1-1→ 𝐵 ∧ 𝐵 ⊆ 𝐶 ) → 𝐹 : 𝐴 –1-1→ 𝐶 )