Theorem relpeq4 44979

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

Assertion

Ref	Expression
relpeq4	⊢ (𝐴 = 𝐶 → (𝐻 RelPres 𝑅, 𝑆(𝐴, 𝐵) ↔ 𝐻 RelPres 𝑅, 𝑆(𝐶, 𝐵)))

Proof of Theorem relpeq4

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

Step	Hyp	Ref	Expression
1		feq2 6630	. . 3 ⊢ (𝐴 = 𝐶 → (𝐻:𝐴⟶𝐵 ↔ 𝐻:𝐶⟶𝐵))
2		raleq 3289	. . . 4 ⊢ (𝐴 = 𝐶 → (∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐻‘𝑥)𝑆(𝐻‘𝑦)) ↔ ∀𝑦 ∈ 𝐶 (𝑥𝑅𝑦 → (𝐻‘𝑥)𝑆(𝐻‘𝑦))))
3	2	raleqbi1dv 3304	. . 3 ⊢ (𝐴 = 𝐶 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐻‘𝑥)𝑆(𝐻‘𝑦)) ↔ ∀𝑥 ∈ 𝐶 ∀𝑦 ∈ 𝐶 (𝑥𝑅𝑦 → (𝐻‘𝑥)𝑆(𝐻‘𝑦))))
4	1, 3	anbi12d 632	. 2 ⊢ (𝐴 = 𝐶 → ((𝐻:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐻‘𝑥)𝑆(𝐻‘𝑦))) ↔ (𝐻:𝐶⟶𝐵 ∧ ∀𝑥 ∈ 𝐶 ∀𝑦 ∈ 𝐶 (𝑥𝑅𝑦 → (𝐻‘𝑥)𝑆(𝐻‘𝑦)))))
5		df-relp 44975	. 2 ⊢ (𝐻 RelPres 𝑅, 𝑆(𝐴, 𝐵) ↔ (𝐻:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐻‘𝑥)𝑆(𝐻‘𝑦))))
6		df-relp 44975	. 2 ⊢ (𝐻 RelPres 𝑅, 𝑆(𝐶, 𝐵) ↔ (𝐻:𝐶⟶𝐵 ∧ ∀𝑥 ∈ 𝐶 ∀𝑦 ∈ 𝐶 (𝑥𝑅𝑦 → (𝐻‘𝑥)𝑆(𝐻‘𝑦))))
7	4, 5, 6	3bitr4g 314	1 ⊢ (𝐴 = 𝐶 → (𝐻 RelPres 𝑅, 𝑆(𝐴, 𝐵) ↔ 𝐻 RelPres 𝑅, 𝑆(𝐶, 𝐵)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541  ∀wral 3047   class class class wbr 5091  ⟶wf 6477  ‘cfv 6481   RelPres wrelp 44974

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1781  df-cleq 2723  df-ral 3048  df-rex 3057  df-fn 6484  df-f 6485  df-relp 44975

This theorem is referenced by: (None)