Theorem relpeq5

Description: Equality theorem for relation-preserving functions. (Contributed by Eric Schmidt, 11-Oct-2025)

	Ref	Expression
Assertion	relpeq5	⊢ ( 𝐵 = 𝐶 → ( 𝐻 RelPres 𝑅 , 𝑆 ( 𝐴 , 𝐵 ) ↔ 𝐻 RelPres 𝑅 , 𝑆 ( 𝐴 , 𝐶 ) ) )

Proof

Step	Hyp	Ref	Expression
1		feq3	⊢ ( 𝐵 = 𝐶 → ( 𝐻 : 𝐴 ⟶ 𝐵 ↔ 𝐻 : 𝐴 ⟶ 𝐶 ) )
2	1	anbi1d	⊢ ( 𝐵 = 𝐶 → ( ( 𝐻 : 𝐴 ⟶ 𝐵 ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝑥 𝑅 𝑦 → ( 𝐻 ‘ 𝑥 ) 𝑆 ( 𝐻 ‘ 𝑦 ) ) ) ↔ ( 𝐻 : 𝐴 ⟶ 𝐶 ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝑥 𝑅 𝑦 → ( 𝐻 ‘ 𝑥 ) 𝑆 ( 𝐻 ‘ 𝑦 ) ) ) ) )
3		df-relp	⊢ ( 𝐻 RelPres 𝑅 , 𝑆 ( 𝐴 , 𝐵 ) ↔ ( 𝐻 : 𝐴 ⟶ 𝐵 ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝑥 𝑅 𝑦 → ( 𝐻 ‘ 𝑥 ) 𝑆 ( 𝐻 ‘ 𝑦 ) ) ) )
4		df-relp	⊢ ( 𝐻 RelPres 𝑅 , 𝑆 ( 𝐴 , 𝐶 ) ↔ ( 𝐻 : 𝐴 ⟶ 𝐶 ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝑥 𝑅 𝑦 → ( 𝐻 ‘ 𝑥 ) 𝑆 ( 𝐻 ‘ 𝑦 ) ) ) )
5	2 3 4	3bitr4g	⊢ ( 𝐵 = 𝐶 → ( 𝐻 RelPres 𝑅 , 𝑆 ( 𝐴 , 𝐵 ) ↔ 𝐻 RelPres 𝑅 , 𝑆 ( 𝐴 , 𝐶 ) ) )