Theorem trclfvss

Description: The transitive closure (as a relation) of a subclass is a subclass of the transitive closure. (Contributed by RP, 3-May-2020)

	Ref	Expression
Assertion	trclfvss	⊢ ( ( 𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊 ∧ 𝑅 ⊆ 𝑆 ) → ( t+ ‘ 𝑅 ) ⊆ ( t+ ‘ 𝑆 ) )

Proof

Step	Hyp	Ref	Expression
1		trclsslem	⊢ ( 𝑅 ⊆ 𝑆 → ∩ { 𝑟 ∣ ( 𝑅 ⊆ 𝑟 ∧ ( 𝑟 ∘ 𝑟 ) ⊆ 𝑟 ) } ⊆ ∩ { 𝑟 ∣ ( 𝑆 ⊆ 𝑟 ∧ ( 𝑟 ∘ 𝑟 ) ⊆ 𝑟 ) } )
2	1	3ad2ant3	⊢ ( ( 𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊 ∧ 𝑅 ⊆ 𝑆 ) → ∩ { 𝑟 ∣ ( 𝑅 ⊆ 𝑟 ∧ ( 𝑟 ∘ 𝑟 ) ⊆ 𝑟 ) } ⊆ ∩ { 𝑟 ∣ ( 𝑆 ⊆ 𝑟 ∧ ( 𝑟 ∘ 𝑟 ) ⊆ 𝑟 ) } )
3		trclfv	⊢ ( 𝑅 ∈ 𝑉 → ( t+ ‘ 𝑅 ) = ∩ { 𝑟 ∣ ( 𝑅 ⊆ 𝑟 ∧ ( 𝑟 ∘ 𝑟 ) ⊆ 𝑟 ) } )
4	3	3ad2ant1	⊢ ( ( 𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊 ∧ 𝑅 ⊆ 𝑆 ) → ( t+ ‘ 𝑅 ) = ∩ { 𝑟 ∣ ( 𝑅 ⊆ 𝑟 ∧ ( 𝑟 ∘ 𝑟 ) ⊆ 𝑟 ) } )
5		trclfv	⊢ ( 𝑆 ∈ 𝑊 → ( t+ ‘ 𝑆 ) = ∩ { 𝑟 ∣ ( 𝑆 ⊆ 𝑟 ∧ ( 𝑟 ∘ 𝑟 ) ⊆ 𝑟 ) } )
6	5	3ad2ant2	⊢ ( ( 𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊 ∧ 𝑅 ⊆ 𝑆 ) → ( t+ ‘ 𝑆 ) = ∩ { 𝑟 ∣ ( 𝑆 ⊆ 𝑟 ∧ ( 𝑟 ∘ 𝑟 ) ⊆ 𝑟 ) } )
7	2 4 6	3sstr4d	⊢ ( ( 𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊 ∧ 𝑅 ⊆ 𝑆 ) → ( t+ ‘ 𝑅 ) ⊆ ( t+ ‘ 𝑆 ) )