Theorem relpf

Description: A relation-preserving function is a function. (Contributed by Eric Schmidt, 11-Oct-2025)

	Ref	Expression
Assertion	relpf	⊢ ( 𝐻 RelPres 𝑅 , 𝑆 ( 𝐴 , 𝐵 ) → 𝐻 : 𝐴 ⟶ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		df-relp	⊢ ( 𝐻 RelPres 𝑅 , 𝑆 ( 𝐴 , 𝐵 ) ↔ ( 𝐻 : 𝐴 ⟶ 𝐵 ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝑥 𝑅 𝑦 → ( 𝐻 ‘ 𝑥 ) 𝑆 ( 𝐻 ‘ 𝑦 ) ) ) )
2	1	simplbi	⊢ ( 𝐻 RelPres 𝑅 , 𝑆 ( 𝐴 , 𝐵 ) → 𝐻 : 𝐴 ⟶ 𝐵 )