Theorem relpeq4 45778

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 6685	. . 3 ⊢ (𝐴 = 𝐶 → (𝐻:𝐴⟶𝐵 ↔ 𝐻:𝐶⟶𝐵))
2		raleq 3318	. . . 4 ⊢ (𝐴 = 𝐶 → (∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐻‘𝑥)𝑆(𝐻‘𝑦)) ↔ ∀𝑦 ∈ 𝐶 (𝑥𝑅𝑦 → (𝐻‘𝑥)𝑆(𝐻‘𝑦))))
3	2	raleqbi1dv 3331	. . 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 644	. 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 45774	. 2 ⊢ (𝐻 RelPres 𝑅, 𝑆(𝐴, 𝐵) ↔ (𝐻:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐻‘𝑥)𝑆(𝐻‘𝑦))))
6		df-relp 45774	. 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 317	1 ⊢ (𝐴 = 𝐶 → (𝐻 RelPres 𝑅, 𝑆(𝐴, 𝐵) ↔ 𝐻 RelPres 𝑅, 𝑆(𝐶, 𝐵)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570  ∀wral 3078   class class class wbr 5107  ⟶wf 6533  ‘cfv 6537   RelPres wrelp 45773

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-9 2155  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-cleq 2754  df-ral 3079  df-rex 3089  df-fn 6540  df-f 6541  df-relp 45774

This theorem is used by: (None)