Theorem predpredss

Description: If A is a subset of B , then their predecessor classes are also subsets. (Contributed by Scott Fenton, 2-Feb-2011)

	Ref	Expression
Assertion	predpredss	⊢ ( 𝐴 ⊆ 𝐵 → Pred ( 𝑅 , 𝐴 , 𝑋 ) ⊆ Pred ( 𝑅 , 𝐵 , 𝑋 ) )

Proof

Step	Hyp	Ref	Expression
1		ssrin	⊢ ( 𝐴 ⊆ 𝐵 → ( 𝐴 ∩ ( ◡ 𝑅 “ { 𝑋 } ) ) ⊆ ( 𝐵 ∩ ( ◡ 𝑅 “ { 𝑋 } ) ) )
2		df-pred	⊢ Pred ( 𝑅 , 𝐴 , 𝑋 ) = ( 𝐴 ∩ ( ◡ 𝑅 “ { 𝑋 } ) )
3		df-pred	⊢ Pred ( 𝑅 , 𝐵 , 𝑋 ) = ( 𝐵 ∩ ( ◡ 𝑅 “ { 𝑋 } ) )
4	1 2 3	3sstr4g	⊢ ( 𝐴 ⊆ 𝐵 → Pred ( 𝑅 , 𝐴 , 𝑋 ) ⊆ Pred ( 𝑅 , 𝐵 , 𝑋 ) )