Theorem clsslem

Description: The closure of a subclass is a subclass of the closure. (Contributed by RP, 16-May-2020)

	Ref	Expression
Assertion	clsslem	⊢ ( 𝑅 ⊆ 𝑆 → ∩ { 𝑟 ∣ ( 𝑅 ⊆ 𝑟 ∧ 𝜑 ) } ⊆ ∩ { 𝑟 ∣ ( 𝑆 ⊆ 𝑟 ∧ 𝜑 ) } )

Proof

Step	Hyp	Ref	Expression
1		sstr2	⊢ ( 𝑅 ⊆ 𝑆 → ( 𝑆 ⊆ 𝑟 → 𝑅 ⊆ 𝑟 ) )
2	1	anim1d	⊢ ( 𝑅 ⊆ 𝑆 → ( ( 𝑆 ⊆ 𝑟 ∧ 𝜑 ) → ( 𝑅 ⊆ 𝑟 ∧ 𝜑 ) ) )
3	2	ss2abdv	⊢ ( 𝑅 ⊆ 𝑆 → { 𝑟 ∣ ( 𝑆 ⊆ 𝑟 ∧ 𝜑 ) } ⊆ { 𝑟 ∣ ( 𝑅 ⊆ 𝑟 ∧ 𝜑 ) } )
4		intss	⊢ ( { 𝑟 ∣ ( 𝑆 ⊆ 𝑟 ∧ 𝜑 ) } ⊆ { 𝑟 ∣ ( 𝑅 ⊆ 𝑟 ∧ 𝜑 ) } → ∩ { 𝑟 ∣ ( 𝑅 ⊆ 𝑟 ∧ 𝜑 ) } ⊆ ∩ { 𝑟 ∣ ( 𝑆 ⊆ 𝑟 ∧ 𝜑 ) } )
5	3 4	syl	⊢ ( 𝑅 ⊆ 𝑆 → ∩ { 𝑟 ∣ ( 𝑅 ⊆ 𝑟 ∧ 𝜑 ) } ⊆ ∩ { 𝑟 ∣ ( 𝑆 ⊆ 𝑟 ∧ 𝜑 ) } )