Theorem fin

Description: Mapping into an intersection. (Contributed by NM, 14-Sep-1999) (Proof shortened by Andrew Salmon, 17-Sep-2011)

	Ref	Expression
Assertion	fin	⊢ ( 𝐹 : 𝐴 ⟶ ( 𝐵 ∩ 𝐶 ) ↔ ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ 𝐹 : 𝐴 ⟶ 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		ssin	⊢ ( ( ran 𝐹 ⊆ 𝐵 ∧ ran 𝐹 ⊆ 𝐶 ) ↔ ran 𝐹 ⊆ ( 𝐵 ∩ 𝐶 ) )
2	1	anbi2i	⊢ ( ( 𝐹 Fn 𝐴 ∧ ( ran 𝐹 ⊆ 𝐵 ∧ ran 𝐹 ⊆ 𝐶 ) ) ↔ ( 𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ ( 𝐵 ∩ 𝐶 ) ) )
3		anandi	⊢ ( ( 𝐹 Fn 𝐴 ∧ ( ran 𝐹 ⊆ 𝐵 ∧ ran 𝐹 ⊆ 𝐶 ) ) ↔ ( ( 𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵 ) ∧ ( 𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐶 ) ) )
4	2 3	bitr3i	⊢ ( ( 𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ ( 𝐵 ∩ 𝐶 ) ) ↔ ( ( 𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵 ) ∧ ( 𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐶 ) ) )
5		df-f	⊢ ( 𝐹 : 𝐴 ⟶ ( 𝐵 ∩ 𝐶 ) ↔ ( 𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ ( 𝐵 ∩ 𝐶 ) ) )
6		df-f	⊢ ( 𝐹 : 𝐴 ⟶ 𝐵 ↔ ( 𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵 ) )
7		df-f	⊢ ( 𝐹 : 𝐴 ⟶ 𝐶 ↔ ( 𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐶 ) )
8	6 7	anbi12i	⊢ ( ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ 𝐹 : 𝐴 ⟶ 𝐶 ) ↔ ( ( 𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵 ) ∧ ( 𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐶 ) ) )
9	4 5 8	3bitr4i	⊢ ( 𝐹 : 𝐴 ⟶ ( 𝐵 ∩ 𝐶 ) ↔ ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ 𝐹 : 𝐴 ⟶ 𝐶 ) )