Theorem imass2

Description: Subset theorem for image. Exercise 22(a) of Enderton p. 53. (Contributed by NM, 22-Mar-1998)

	Ref	Expression
Assertion	imass2	⊢ ( 𝐴 ⊆ 𝐵 → ( 𝐶 “ 𝐴 ) ⊆ ( 𝐶 “ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		ssres2	⊢ ( 𝐴 ⊆ 𝐵 → ( 𝐶 ↾ 𝐴 ) ⊆ ( 𝐶 ↾ 𝐵 ) )
2		rnss	⊢ ( ( 𝐶 ↾ 𝐴 ) ⊆ ( 𝐶 ↾ 𝐵 ) → ran ( 𝐶 ↾ 𝐴 ) ⊆ ran ( 𝐶 ↾ 𝐵 ) )
3	1 2	syl	⊢ ( 𝐴 ⊆ 𝐵 → ran ( 𝐶 ↾ 𝐴 ) ⊆ ran ( 𝐶 ↾ 𝐵 ) )
4		df-ima	⊢ ( 𝐶 “ 𝐴 ) = ran ( 𝐶 ↾ 𝐴 )
5		df-ima	⊢ ( 𝐶 “ 𝐵 ) = ran ( 𝐶 ↾ 𝐵 )
6	3 4 5	3sstr4g	⊢ ( 𝐴 ⊆ 𝐵 → ( 𝐶 “ 𝐴 ) ⊆ ( 𝐶 “ 𝐵 ) )