Theorem relpeq1 45294

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

Assertion

Ref	Expression
relpeq1	⊢ (𝐻 = 𝐺 → (𝐻 RelPres 𝑅, 𝑆(𝐴, 𝐵) ↔ 𝐺 RelPres 𝑅, 𝑆(𝐴, 𝐵)))

Proof of Theorem relpeq1

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

Step	Hyp	Ref	Expression
1		feq1 6648	. . 3 ⊢ (𝐻 = 𝐺 → (𝐻:𝐴⟶𝐵 ↔ 𝐺:𝐴⟶𝐵))
2		fveq1 6841	. . . . . 6 ⊢ (𝐻 = 𝐺 → (𝐻‘𝑥) = (𝐺‘𝑥))
3		fveq1 6841	. . . . . 6 ⊢ (𝐻 = 𝐺 → (𝐻‘𝑦) = (𝐺‘𝑦))
4	2, 3	breq12d 5113	. . . . 5 ⊢ (𝐻 = 𝐺 → ((𝐻‘𝑥)𝑆(𝐻‘𝑦) ↔ (𝐺‘𝑥)𝑆(𝐺‘𝑦)))
5	4	imbi2d 340	. . . 4 ⊢ (𝐻 = 𝐺 → ((𝑥𝑅𝑦 → (𝐻‘𝑥)𝑆(𝐻‘𝑦)) ↔ (𝑥𝑅𝑦 → (𝐺‘𝑥)𝑆(𝐺‘𝑦))))
6	5	2ralbidv 3202	. . 3 ⊢ (𝐻 = 𝐺 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐻‘𝑥)𝑆(𝐻‘𝑦)) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐺‘𝑥)𝑆(𝐺‘𝑦))))
7	1, 6	anbi12d 633	. 2 ⊢ (𝐻 = 𝐺 → ((𝐻:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐻‘𝑥)𝑆(𝐻‘𝑦))) ↔ (𝐺:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐺‘𝑥)𝑆(𝐺‘𝑦)))))
8		df-relp 45293	. 2 ⊢ (𝐻 RelPres 𝑅, 𝑆(𝐴, 𝐵) ↔ (𝐻:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐻‘𝑥)𝑆(𝐻‘𝑦))))
9		df-relp 45293	. 2 ⊢ (𝐺 RelPres 𝑅, 𝑆(𝐴, 𝐵) ↔ (𝐺:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐺‘𝑥)𝑆(𝐺‘𝑦))))
10	7, 8, 9	3bitr4g 314	1 ⊢ (𝐻 = 𝐺 → (𝐻 RelPres 𝑅, 𝑆(𝐴, 𝐵) ↔ 𝐺 RelPres 𝑅, 𝑆(𝐴, 𝐵)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542  ∀wral 3052   class class class wbr 5100  ⟶wf 6496  ‘cfv 6500   RelPres wrelp 45292

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-fv 6508  df-relp 45293

This theorem is referenced by: (None)