Theorem xphe

Description: Any Cartesian product is hereditary in its second class. (Contributed by RP, 27-Mar-2020) (Proof shortened by OpenAI, 3-Jul-2020)

	Ref	Expression
Assertion	xphe	⊢ ( 𝐴 × 𝐵 ) hereditary 𝐵

Proof

Step	Hyp	Ref	Expression
1		imassrn	⊢ ( ( 𝐴 × 𝐵 ) “ 𝐵 ) ⊆ ran ( 𝐴 × 𝐵 )
2		rnxpss	⊢ ran ( 𝐴 × 𝐵 ) ⊆ 𝐵
3	1 2	sstri	⊢ ( ( 𝐴 × 𝐵 ) “ 𝐵 ) ⊆ 𝐵
4		df-he	⊢ ( ( 𝐴 × 𝐵 ) hereditary 𝐵 ↔ ( ( 𝐴 × 𝐵 ) “ 𝐵 ) ⊆ 𝐵 )
5	3 4	mpbir	⊢ ( 𝐴 × 𝐵 ) hereditary 𝐵