Theorem imass1

Description: Subset theorem for image. (Contributed by NM, 16-Mar-2004)

	Ref	Expression
Assertion	imass1	⊢ ( 𝐴 ⊆ 𝐵 → ( 𝐴 “ 𝐶 ) ⊆ ( 𝐵 “ 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		ssres	⊢ ( 𝐴 ⊆ 𝐵 → ( 𝐴 ↾ 𝐶 ) ⊆ ( 𝐵 ↾ 𝐶 ) )
2		rnss	⊢ ( ( 𝐴 ↾ 𝐶 ) ⊆ ( 𝐵 ↾ 𝐶 ) → ran ( 𝐴 ↾ 𝐶 ) ⊆ ran ( 𝐵 ↾ 𝐶 ) )
3	1 2	syl	⊢ ( 𝐴 ⊆ 𝐵 → ran ( 𝐴 ↾ 𝐶 ) ⊆ ran ( 𝐵 ↾ 𝐶 ) )
4		df-ima	⊢ ( 𝐴 “ 𝐶 ) = ran ( 𝐴 ↾ 𝐶 )
5		df-ima	⊢ ( 𝐵 “ 𝐶 ) = ran ( 𝐵 ↾ 𝐶 )
6	3 4 5	3sstr4g	⊢ ( 𝐴 ⊆ 𝐵 → ( 𝐴 “ 𝐶 ) ⊆ ( 𝐵 “ 𝐶 ) )