Theorem fss

Description: Expanding the codomain of a mapping. (Contributed by NM, 10-May-1998) (Proof shortened by Andrew Salmon, 17-Sep-2011)

	Ref	Expression
Assertion	fss	⊢ ( ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ 𝐵 ⊆ 𝐶 ) → 𝐹 : 𝐴 ⟶ 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		sstr2	⊢ ( ran 𝐹 ⊆ 𝐵 → ( 𝐵 ⊆ 𝐶 → ran 𝐹 ⊆ 𝐶 ) )
2	1	com12	⊢ ( 𝐵 ⊆ 𝐶 → ( ran 𝐹 ⊆ 𝐵 → ran 𝐹 ⊆ 𝐶 ) )
3	2	anim2d	⊢ ( 𝐵 ⊆ 𝐶 → ( ( 𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵 ) → ( 𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐶 ) ) )
4		df-f	⊢ ( 𝐹 : 𝐴 ⟶ 𝐵 ↔ ( 𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵 ) )
5		df-f	⊢ ( 𝐹 : 𝐴 ⟶ 𝐶 ↔ ( 𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐶 ) )
6	3 4 5	3imtr4g	⊢ ( 𝐵 ⊆ 𝐶 → ( 𝐹 : 𝐴 ⟶ 𝐵 → 𝐹 : 𝐴 ⟶ 𝐶 ) )
7	6	impcom	⊢ ( ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ 𝐵 ⊆ 𝐶 ) → 𝐹 : 𝐴 ⟶ 𝐶 )