Theorem chnrin2

Description: Distribution of chain class constructor over relation intersection. (Contributed by Ender Ting, 24-Jul-2026)

	Ref	Expression
Assertion	chnrin2	$$⊢{\mathrm{Chain}}_B\left(R\cap \underset{\dot{}}{<}\right)={\mathrm{Chain}}_{B}R\cap {\mathrm{Chain}}_B\underset{\dot{}}{<}$$

Proof

Step	Hyp	Ref	Expression
1		chnrrin	$$⊢n\in {\mathrm{Chain}}_B\left(R\cap \underset{\dot{}}{<}\right)\to \left(n\in {\mathrm{Chain}}_{B}R\wedge n\in {\mathrm{Chain}}_B\underset{\dot{}}{<}\right)$$
2		chnrin	$$⊢\left(n\in {\mathrm{Chain}}_{B}R\wedge n\in {\mathrm{Chain}}_B\underset{\dot{}}{<}\right)\to n\in {\mathrm{Chain}}_B\left(R\cap \underset{\dot{}}{<}\right)$$
3	1 2	impbii	$$⊢n\in {\mathrm{Chain}}_B\left(R\cap \underset{\dot{}}{<}\right)\leftrightarrow \left(n\in {\mathrm{Chain}}_{B}R\wedge n\in {\mathrm{Chain}}_B\underset{\dot{}}{<}\right)$$
4		elin	$$⊢n\in \left({\mathrm{Chain}}_{B}R\cap {\mathrm{Chain}}_B\underset{\dot{}}{<}\right)\leftrightarrow \left(n\in {\mathrm{Chain}}_{B}R\wedge n\in {\mathrm{Chain}}_B\underset{\dot{}}{<}\right)$$
5	3 4	bitr4i	$$⊢n\in {\mathrm{Chain}}_B\left(R\cap \underset{\dot{}}{<}\right)\leftrightarrow n\in \left({\mathrm{Chain}}_{B}R\cap {\mathrm{Chain}}_B\underset{\dot{}}{<}\right)$$
6	5	eqriv	$$⊢{\mathrm{Chain}}_B\left(R\cap \underset{\dot{}}{<}\right)={\mathrm{Chain}}_{B}R\cap {\mathrm{Chain}}_B\underset{\dot{}}{<}$$