Theorem f1resf1

Description: The restriction of an injective function is injective. (Contributed by AV, 28-Jun-2022)

	Ref	Expression
Assertion	f1resf1	⊢ ( ( 𝐹 : 𝐴 –1-1→ 𝐵 ∧ 𝐶 ⊆ 𝐴 ∧ ( 𝐹 ↾ 𝐶 ) : 𝐶 ⟶ 𝐷 ) → ( 𝐹 ↾ 𝐶 ) : 𝐶 –1-1→ 𝐷 )

Proof

Step	Hyp	Ref	Expression
1		f1ssres	⊢ ( ( 𝐹 : 𝐴 –1-1→ 𝐵 ∧ 𝐶 ⊆ 𝐴 ) → ( 𝐹 ↾ 𝐶 ) : 𝐶 –1-1→ 𝐵 )
2	1	3adant3	⊢ ( ( 𝐹 : 𝐴 –1-1→ 𝐵 ∧ 𝐶 ⊆ 𝐴 ∧ ( 𝐹 ↾ 𝐶 ) : 𝐶 ⟶ 𝐷 ) → ( 𝐹 ↾ 𝐶 ) : 𝐶 –1-1→ 𝐵 )
3		frn	⊢ ( ( 𝐹 ↾ 𝐶 ) : 𝐶 ⟶ 𝐷 → ran ( 𝐹 ↾ 𝐶 ) ⊆ 𝐷 )
4	3	3ad2ant3	⊢ ( ( 𝐹 : 𝐴 –1-1→ 𝐵 ∧ 𝐶 ⊆ 𝐴 ∧ ( 𝐹 ↾ 𝐶 ) : 𝐶 ⟶ 𝐷 ) → ran ( 𝐹 ↾ 𝐶 ) ⊆ 𝐷 )
5		f1ssr	⊢ ( ( ( 𝐹 ↾ 𝐶 ) : 𝐶 –1-1→ 𝐵 ∧ ran ( 𝐹 ↾ 𝐶 ) ⊆ 𝐷 ) → ( 𝐹 ↾ 𝐶 ) : 𝐶 –1-1→ 𝐷 )
6	2 4 5	syl2anc	⊢ ( ( 𝐹 : 𝐴 –1-1→ 𝐵 ∧ 𝐶 ⊆ 𝐴 ∧ ( 𝐹 ↾ 𝐶 ) : 𝐶 ⟶ 𝐷 ) → ( 𝐹 ↾ 𝐶 ) : 𝐶 –1-1→ 𝐷 )