Theorem relpeq2 46077

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

Assertion

Ref	Expression
relpeq2	⊢ (𝑅 = 𝑇 → (𝐻 RelPres 𝑅, 𝑆(𝐴, 𝐵) ↔ 𝐻 RelPres 𝑇, 𝑆(𝐴, 𝐵)))

Proof of Theorem relpeq2

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		breq 5108	. . . . 5 ⊢ (𝑅 = 𝑇 → (𝑥𝑅𝑦 ↔ 𝑥𝑇𝑦))
2	1	imbi1d 344	. . . 4 ⊢ (𝑅 = 𝑇 → ((𝑥𝑅𝑦 → (𝐻‘𝑥)𝑆(𝐻‘𝑦)) ↔ (𝑥𝑇𝑦 → (𝐻‘𝑥)𝑆(𝐻‘𝑦))))
3	2	2ralbidv 3230	. . 3 ⊢ (𝑅 = 𝑇 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐻‘𝑥)𝑆(𝐻‘𝑦)) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑇𝑦 → (𝐻‘𝑥)𝑆(𝐻‘𝑦))))
4	3	anbi2d 642	. 2 ⊢ (𝑅 = 𝑇 → ((𝐻:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐻‘𝑥)𝑆(𝐻‘𝑦))) ↔ (𝐻:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑇𝑦 → (𝐻‘𝑥)𝑆(𝐻‘𝑦)))))
5		df-relp 46075	. 2 ⊢ (𝐻 RelPres 𝑅, 𝑆(𝐴, 𝐵) ↔ (𝐻:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐻‘𝑥)𝑆(𝐻‘𝑦))))
6		df-relp 46075	. 2 ⊢ (𝐻 RelPres 𝑇, 𝑆(𝐴, 𝐵) ↔ (𝐻:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑇𝑦 → (𝐻‘𝑥)𝑆(𝐻‘𝑦))))
7	4, 5, 6	3bitr4g 317	1 ⊢ (𝑅 = 𝑇 → (𝐻 RelPres 𝑅, 𝑆(𝐴, 𝐵) ↔ 𝐻 RelPres 𝑇, 𝑆(𝐴, 𝐵)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569  ∀wral 3080   class class class wbr 5106  ⟶wf 6543  ‘cfv 6547   RelPres wrelp 46074

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1812  df-cleq 2755  df-clel 2839  df-ral 3081  df-br 5107  df-relp 46075

This theorem is used by: (None)