Theorem rnss

Description: Subset theorem for range. (Contributed by NM, 22-Mar-1998)

	Ref	Expression
Assertion	rnss	⊢ ( 𝐴 ⊆ 𝐵 → ran 𝐴 ⊆ ran 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		cnvss	⊢ ( 𝐴 ⊆ 𝐵 → ◡ 𝐴 ⊆ ◡ 𝐵 )
2		dmss	⊢ ( ◡ 𝐴 ⊆ ◡ 𝐵 → dom ◡ 𝐴 ⊆ dom ◡ 𝐵 )
3	1 2	syl	⊢ ( 𝐴 ⊆ 𝐵 → dom ◡ 𝐴 ⊆ dom ◡ 𝐵 )
4		df-rn	⊢ ran 𝐴 = dom ◡ 𝐴
5		df-rn	⊢ ran 𝐵 = dom ◡ 𝐵
6	3 4 5	3sstr4g	⊢ ( 𝐴 ⊆ 𝐵 → ran 𝐴 ⊆ ran 𝐵 )