Theorem parteq2

Description: Equality theorem for partition. (Contributed by Peter Mazsa, 25-Jul-2024)

	Ref	Expression
Assertion	parteq2	$$⊢A=B\to \left(RPartA\leftrightarrow RPartB\right)$$

Proof

Step	Hyp	Ref	Expression
1		eqeq2	$$⊢A=B\to \left(\mathrm{dom}R/R=A\leftrightarrow \mathrm{dom}R/R=B\right)$$
2	1	anbi2d	$$⊢A=B\to \left(\left(DisjR\wedge \mathrm{dom}R/R=A\right)\leftrightarrow \left(DisjR\wedge \mathrm{dom}R/R=B\right)\right)$$
3		dfpart2	$$⊢RPartA\leftrightarrow \left(DisjR\wedge \mathrm{dom}R/R=A\right)$$
4		dfpart2	$$⊢RPartB\leftrightarrow \left(DisjR\wedge \mathrm{dom}R/R=B\right)$$
5	2 3 4	3bitr4g	$$⊢A=B\to \left(RPartA\leftrightarrow RPartB\right)$$