Theorem relprel

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

	Ref	Expression
Assertion	relprel	⊢ ( ( 𝐻 RelPres 𝑅 , 𝑆 ( 𝐴 , 𝐵 ) ∧ ( 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴 ) ) → ( 𝐶 𝑅 𝐷 → ( 𝐻 ‘ 𝐶 ) 𝑆 ( 𝐻 ‘ 𝐷 ) ) )

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	simprbi	⊢ ( 𝐻 RelPres 𝑅 , 𝑆 ( 𝐴 , 𝐵 ) → ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝑥 𝑅 𝑦 → ( 𝐻 ‘ 𝑥 ) 𝑆 ( 𝐻 ‘ 𝑦 ) ) )
3		breq1	⊢ ( 𝑥 = 𝐶 → ( 𝑥 𝑅 𝑦 ↔ 𝐶 𝑅 𝑦 ) )
4		fveq2	⊢ ( 𝑥 = 𝐶 → ( 𝐻 ‘ 𝑥 ) = ( 𝐻 ‘ 𝐶 ) )
5	4	breq1d	⊢ ( 𝑥 = 𝐶 → ( ( 𝐻 ‘ 𝑥 ) 𝑆 ( 𝐻 ‘ 𝑦 ) ↔ ( 𝐻 ‘ 𝐶 ) 𝑆 ( 𝐻 ‘ 𝑦 ) ) )
6	3 5	imbi12d	⊢ ( 𝑥 = 𝐶 → ( ( 𝑥 𝑅 𝑦 → ( 𝐻 ‘ 𝑥 ) 𝑆 ( 𝐻 ‘ 𝑦 ) ) ↔ ( 𝐶 𝑅 𝑦 → ( 𝐻 ‘ 𝐶 ) 𝑆 ( 𝐻 ‘ 𝑦 ) ) ) )
7		breq2	⊢ ( 𝑦 = 𝐷 → ( 𝐶 𝑅 𝑦 ↔ 𝐶 𝑅 𝐷 ) )
8		fveq2	⊢ ( 𝑦 = 𝐷 → ( 𝐻 ‘ 𝑦 ) = ( 𝐻 ‘ 𝐷 ) )
9	8	breq2d	⊢ ( 𝑦 = 𝐷 → ( ( 𝐻 ‘ 𝐶 ) 𝑆 ( 𝐻 ‘ 𝑦 ) ↔ ( 𝐻 ‘ 𝐶 ) 𝑆 ( 𝐻 ‘ 𝐷 ) ) )
10	7 9	imbi12d	⊢ ( 𝑦 = 𝐷 → ( ( 𝐶 𝑅 𝑦 → ( 𝐻 ‘ 𝐶 ) 𝑆 ( 𝐻 ‘ 𝑦 ) ) ↔ ( 𝐶 𝑅 𝐷 → ( 𝐻 ‘ 𝐶 ) 𝑆 ( 𝐻 ‘ 𝐷 ) ) ) )
11	6 10	rspc2v	⊢ ( ( 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴 ) → ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝑥 𝑅 𝑦 → ( 𝐻 ‘ 𝑥 ) 𝑆 ( 𝐻 ‘ 𝑦 ) ) → ( 𝐶 𝑅 𝐷 → ( 𝐻 ‘ 𝐶 ) 𝑆 ( 𝐻 ‘ 𝐷 ) ) ) )
12	2 11	mpan9	⊢ ( ( 𝐻 RelPres 𝑅 , 𝑆 ( 𝐴 , 𝐵 ) ∧ ( 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴 ) ) → ( 𝐶 𝑅 𝐷 → ( 𝐻 ‘ 𝐶 ) 𝑆 ( 𝐻 ‘ 𝐷 ) ) )