Theorem hess

Description: Subclass law for relations being herditary over a class. (Contributed by RP, 27-Mar-2020)

	Ref	Expression
Assertion	hess	⊢ ( 𝑆 ⊆ 𝑅 → ( 𝑅 hereditary 𝐴 → 𝑆 hereditary 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		imass1	⊢ ( 𝑆 ⊆ 𝑅 → ( 𝑆 “ 𝐴 ) ⊆ ( 𝑅 “ 𝐴 ) )
2		sstr2	⊢ ( ( 𝑆 “ 𝐴 ) ⊆ ( 𝑅 “ 𝐴 ) → ( ( 𝑅 “ 𝐴 ) ⊆ 𝐴 → ( 𝑆 “ 𝐴 ) ⊆ 𝐴 ) )
3	1 2	syl	⊢ ( 𝑆 ⊆ 𝑅 → ( ( 𝑅 “ 𝐴 ) ⊆ 𝐴 → ( 𝑆 “ 𝐴 ) ⊆ 𝐴 ) )
4		df-he	⊢ ( 𝑅 hereditary 𝐴 ↔ ( 𝑅 “ 𝐴 ) ⊆ 𝐴 )
5		df-he	⊢ ( 𝑆 hereditary 𝐴 ↔ ( 𝑆 “ 𝐴 ) ⊆ 𝐴 )
6	3 4 5	3imtr4g	⊢ ( 𝑆 ⊆ 𝑅 → ( 𝑅 hereditary 𝐴 → 𝑆 hereditary 𝐴 ) )