Theorem parteq1

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

	Ref	Expression
Assertion	parteq1	⊢ ( 𝑅 = 𝑆 → ( 𝑅 Part 𝐴 ↔ 𝑆 Part 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		disjdmqseqeq1	⊢ ( 𝑅 = 𝑆 → ( ( Disj 𝑅 ∧ ( dom 𝑅 / 𝑅 ) = 𝐴 ) ↔ ( Disj 𝑆 ∧ ( dom 𝑆 / 𝑆 ) = 𝐴 ) ) )
2		dfpart2	⊢ ( 𝑅 Part 𝐴 ↔ ( Disj 𝑅 ∧ ( dom 𝑅 / 𝑅 ) = 𝐴 ) )
3		dfpart2	⊢ ( 𝑆 Part 𝐴 ↔ ( Disj 𝑆 ∧ ( dom 𝑆 / 𝑆 ) = 𝐴 ) )
4	1 2 3	3bitr4g	⊢ ( 𝑅 = 𝑆 → ( 𝑅 Part 𝐴 ↔ 𝑆 Part 𝐴 ) )