Theorem relpeq3

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

	Ref	Expression
Assertion	relpeq3	⊢ ( 𝑆 = 𝑇 → ( 𝐻 RelPres 𝑅 , 𝑆 ( 𝐴 , 𝐵 ) ↔ 𝐻 RelPres 𝑅 , 𝑇 ( 𝐴 , 𝐵 ) ) )

Proof

Step	Hyp	Ref	Expression
1		breq	⊢ ( 𝑆 = 𝑇 → ( ( 𝐻 ‘ 𝑥 ) 𝑆 ( 𝐻 ‘ 𝑦 ) ↔ ( 𝐻 ‘ 𝑥 ) 𝑇 ( 𝐻 ‘ 𝑦 ) ) )
2	1	imbi2d	⊢ ( 𝑆 = 𝑇 → ( ( 𝑥 𝑅 𝑦 → ( 𝐻 ‘ 𝑥 ) 𝑆 ( 𝐻 ‘ 𝑦 ) ) ↔ ( 𝑥 𝑅 𝑦 → ( 𝐻 ‘ 𝑥 ) 𝑇 ( 𝐻 ‘ 𝑦 ) ) ) )
3	2	2ralbidv	⊢ ( 𝑆 = 𝑇 → ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝑥 𝑅 𝑦 → ( 𝐻 ‘ 𝑥 ) 𝑆 ( 𝐻 ‘ 𝑦 ) ) ↔ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝑥 𝑅 𝑦 → ( 𝐻 ‘ 𝑥 ) 𝑇 ( 𝐻 ‘ 𝑦 ) ) ) )
4	3	anbi2d	⊢ ( 𝑆 = 𝑇 → ( ( 𝐻 : 𝐴 ⟶ 𝐵 ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝑥 𝑅 𝑦 → ( 𝐻 ‘ 𝑥 ) 𝑆 ( 𝐻 ‘ 𝑦 ) ) ) ↔ ( 𝐻 : 𝐴 ⟶ 𝐵 ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝑥 𝑅 𝑦 → ( 𝐻 ‘ 𝑥 ) 𝑇 ( 𝐻 ‘ 𝑦 ) ) ) ) )
5		df-relp	⊢ ( 𝐻 RelPres 𝑅 , 𝑆 ( 𝐴 , 𝐵 ) ↔ ( 𝐻 : 𝐴 ⟶ 𝐵 ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝑥 𝑅 𝑦 → ( 𝐻 ‘ 𝑥 ) 𝑆 ( 𝐻 ‘ 𝑦 ) ) ) )
6		df-relp	⊢ ( 𝐻 RelPres 𝑅 , 𝑇 ( 𝐴 , 𝐵 ) ↔ ( 𝐻 : 𝐴 ⟶ 𝐵 ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝑥 𝑅 𝑦 → ( 𝐻 ‘ 𝑥 ) 𝑇 ( 𝐻 ‘ 𝑦 ) ) ) )
7	4 5 6	3bitr4g	⊢ ( 𝑆 = 𝑇 → ( 𝐻 RelPres 𝑅 , 𝑆 ( 𝐴 , 𝐵 ) ↔ 𝐻 RelPres 𝑅 , 𝑇 ( 𝐴 , 𝐵 ) ) )