Theorem relpeq5 45522

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

Assertion

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

Proof of Theorem relpeq5

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

Step	Hyp	Ref	Expression
1		feq3 6675	. . 3 ⊢ (𝐵 = 𝐶 → (𝐻:𝐴⟶𝐵 ↔ 𝐻:𝐴⟶𝐶))
2	1	anbi1d 642	. 2 ⊢ (𝐵 = 𝐶 → ((𝐻:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐻‘𝑥)𝑆(𝐻‘𝑦))) ↔ (𝐻:𝐴⟶𝐶 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐻‘𝑥)𝑆(𝐻‘𝑦)))))
3		df-relp 45517	. 2 ⊢ (𝐻 RelPres 𝑅, 𝑆(𝐴, 𝐵) ↔ (𝐻:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐻‘𝑥)𝑆(𝐻‘𝑦))))
4		df-relp 45517	. 2 ⊢ (𝐻 RelPres 𝑅, 𝑆(𝐴, 𝐶) ↔ (𝐻:𝐴⟶𝐶 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐻‘𝑥)𝑆(𝐻‘𝑦))))
5	2, 3, 4	3bitr4g 317	1 ⊢ (𝐵 = 𝐶 → (𝐻 RelPres 𝑅, 𝑆(𝐴, 𝐵) ↔ 𝐻 RelPres 𝑅, 𝑆(𝐴, 𝐶)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1563  ∀wral 3079   class class class wbr 5105  ⟶wf 6521  ‘cfv 6525   RelPres wrelp 45516

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-9 2155  ax-ext 2737

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1803  df-cleq 2757  df-ss 3924  df-f 6529  df-relp 45517

This theorem is referenced by: (None)