Theorem parteq2

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

	Ref	Expression
Assertion	parteq2	⊢ ( 𝐴 = 𝐵 → ( 𝑅 Part 𝐴 ↔ 𝑅 Part 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		eqeq2	⊢ ( 𝐴 = 𝐵 → ( ( dom 𝑅 / 𝑅 ) = 𝐴 ↔ ( dom 𝑅 / 𝑅 ) = 𝐵 ) )
2	1	anbi2d	⊢ ( 𝐴 = 𝐵 → ( ( Disj 𝑅 ∧ ( dom 𝑅 / 𝑅 ) = 𝐴 ) ↔ ( Disj 𝑅 ∧ ( dom 𝑅 / 𝑅 ) = 𝐵 ) ) )
3		dfpart2	⊢ ( 𝑅 Part 𝐴 ↔ ( Disj 𝑅 ∧ ( dom 𝑅 / 𝑅 ) = 𝐴 ) )
4		dfpart2	⊢ ( 𝑅 Part 𝐵 ↔ ( Disj 𝑅 ∧ ( dom 𝑅 / 𝑅 ) = 𝐵 ) )
5	2 3 4	3bitr4g	⊢ ( 𝐴 = 𝐵 → ( 𝑅 Part 𝐴 ↔ 𝑅 Part 𝐵 ) )