Theorem predrelss

Description: Subset carries from relation to predecessor class. (Contributed by Scott Fenton, 25-Nov-2024)

	Ref	Expression
Assertion	predrelss	⊢ ( 𝑅 ⊆ 𝑆 → Pred ( 𝑅 , 𝐴 , 𝑋 ) ⊆ Pred ( 𝑆 , 𝐴 , 𝑋 ) )

Proof

Step	Hyp	Ref	Expression
1		cnvss	⊢ ( 𝑅 ⊆ 𝑆 → ◡ 𝑅 ⊆ ◡ 𝑆 )
2		imass1	⊢ ( ◡ 𝑅 ⊆ ◡ 𝑆 → ( ◡ 𝑅 “ { 𝑋 } ) ⊆ ( ◡ 𝑆 “ { 𝑋 } ) )
3		sslin	⊢ ( ( ◡ 𝑅 “ { 𝑋 } ) ⊆ ( ◡ 𝑆 “ { 𝑋 } ) → ( 𝐴 ∩ ( ◡ 𝑅 “ { 𝑋 } ) ) ⊆ ( 𝐴 ∩ ( ◡ 𝑆 “ { 𝑋 } ) ) )
4	1 2 3	3syl	⊢ ( 𝑅 ⊆ 𝑆 → ( 𝐴 ∩ ( ◡ 𝑅 “ { 𝑋 } ) ) ⊆ ( 𝐴 ∩ ( ◡ 𝑆 “ { 𝑋 } ) ) )
5		df-pred	⊢ Pred ( 𝑅 , 𝐴 , 𝑋 ) = ( 𝐴 ∩ ( ◡ 𝑅 “ { 𝑋 } ) )
6		df-pred	⊢ Pred ( 𝑆 , 𝐴 , 𝑋 ) = ( 𝐴 ∩ ( ◡ 𝑆 “ { 𝑋 } ) )
7	4 5 6	3sstr4g	⊢ ( 𝑅 ⊆ 𝑆 → Pred ( 𝑅 , 𝐴 , 𝑋 ) ⊆ Pred ( 𝑆 , 𝐴 , 𝑋 ) )