Theorem heeq12

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

	Ref	Expression
Assertion	heeq12	$$⊢\left(R=S\wedge A=B\right)\to \left(RhereditaryA\leftrightarrow ShereditaryB\right)$$

Proof

Step	Hyp	Ref	Expression
1		simpl	$$⊢\left(R=S\wedge A=B\right)\to R=S$$
2		simpr	$$⊢\left(R=S\wedge A=B\right)\to A=B$$
3	1 2	imaeq12d	$$⊢\left(R=S\wedge A=B\right)\to R\left[A\right]=S\left[B\right]$$
4	3 2	sseq12d	$$⊢\left(R=S\wedge A=B\right)\to \left(R\left[A\right]\subseteq A\leftrightarrow S\left[B\right]\subseteq B\right)$$
5		df-he	$$⊢RhereditaryA\leftrightarrow R\left[A\right]\subseteq A$$
6		df-he	$$⊢ShereditaryB\leftrightarrow S\left[B\right]\subseteq B$$
7	4 5 6	3bitr4g	$$⊢\left(R=S\wedge A=B\right)\to \left(RhereditaryA\leftrightarrow ShereditaryB\right)$$