Theorem refreleq

Description: Equality theorem for reflexive relation. (Contributed by Peter Mazsa, 15-Apr-2019) (Revised by Peter Mazsa, 23-Sep-2021)

	Ref	Expression
Assertion	refreleq	⊢ ( 𝑅 = 𝑆 → ( RefRel 𝑅 ↔ RefRel 𝑆 ) )

Proof

Step	Hyp	Ref	Expression
1		dmeq	⊢ ( 𝑅 = 𝑆 → dom 𝑅 = dom 𝑆 )
2		rneq	⊢ ( 𝑅 = 𝑆 → ran 𝑅 = ran 𝑆 )
3	1 2	xpeq12d	⊢ ( 𝑅 = 𝑆 → ( dom 𝑅 × ran 𝑅 ) = ( dom 𝑆 × ran 𝑆 ) )
4	3	ineq2d	⊢ ( 𝑅 = 𝑆 → ( I ∩ ( dom 𝑅 × ran 𝑅 ) ) = ( I ∩ ( dom 𝑆 × ran 𝑆 ) ) )
5		id	⊢ ( 𝑅 = 𝑆 → 𝑅 = 𝑆 )
6	4 5	sseq12d	⊢ ( 𝑅 = 𝑆 → ( ( I ∩ ( dom 𝑅 × ran 𝑅 ) ) ⊆ 𝑅 ↔ ( I ∩ ( dom 𝑆 × ran 𝑆 ) ) ⊆ 𝑆 ) )
7		releq	⊢ ( 𝑅 = 𝑆 → ( Rel 𝑅 ↔ Rel 𝑆 ) )
8	6 7	anbi12d	⊢ ( 𝑅 = 𝑆 → ( ( ( I ∩ ( dom 𝑅 × ran 𝑅 ) ) ⊆ 𝑅 ∧ Rel 𝑅 ) ↔ ( ( I ∩ ( dom 𝑆 × ran 𝑆 ) ) ⊆ 𝑆 ∧ Rel 𝑆 ) ) )
9		dfrefrel2	⊢ ( RefRel 𝑅 ↔ ( ( I ∩ ( dom 𝑅 × ran 𝑅 ) ) ⊆ 𝑅 ∧ Rel 𝑅 ) )
10		dfrefrel2	⊢ ( RefRel 𝑆 ↔ ( ( I ∩ ( dom 𝑆 × ran 𝑆 ) ) ⊆ 𝑆 ∧ Rel 𝑆 ) )
11	8 9 10	3bitr4g	⊢ ( 𝑅 = 𝑆 → ( RefRel 𝑅 ↔ RefRel 𝑆 ) )