Theorem parteq1

Description: Equality theorem for partition. (Contributed by Peter Mazsa, 5-Oct-2021)

	Ref	Expression
Assertion	parteq1	$$⊢R=S\to \left(RPartA\leftrightarrow SPartA\right)$$

Proof

Step	Hyp	Ref	Expression
1		disjdmqseqeq1	$$⊢R=S\to \left(\left(DisjR\wedge \mathrm{dom}R/R=A\right)\leftrightarrow \left(DisjS\wedge \mathrm{dom}S/S=A\right)\right)$$
2		dfpart2	$$⊢RPartA\leftrightarrow \left(DisjR\wedge \mathrm{dom}R/R=A\right)$$
3		dfpart2	$$⊢SPartA\leftrightarrow \left(DisjS\wedge \mathrm{dom}S/S=A\right)$$
4	1 2 3	3bitr4g	$$⊢R=S\to \left(RPartA\leftrightarrow SPartA\right)$$