Theorem resimass

Description: The image of a restriction is a subset of the original image. (Contributed by Glauco Siliprandi, 23-Oct-2021)

	Ref	Expression
Assertion	resimass	⊢ ( ( 𝐴 ↾ 𝐵 ) “ 𝐶 ) ⊆ ( 𝐴 “ 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		resss	⊢ ( 𝐴 ↾ 𝐵 ) ⊆ 𝐴
2		imass1	⊢ ( ( 𝐴 ↾ 𝐵 ) ⊆ 𝐴 → ( ( 𝐴 ↾ 𝐵 ) “ 𝐶 ) ⊆ ( 𝐴 “ 𝐶 ) )
3	1 2	ax-mp	⊢ ( ( 𝐴 ↾ 𝐵 ) “ 𝐶 ) ⊆ ( 𝐴 “ 𝐶 )