Theorem isoeq3 7327

Description: Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.)

Assertion

Ref	Expression
isoeq3	⊢ (𝑆 = 𝑇 → (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐻 Isom 𝑅, 𝑇 (𝐴, 𝐵)))

Proof of Theorem isoeq3

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

Step	Hyp	Ref	Expression
1		breq 5150	. . . . 5 ⊢ (𝑆 = 𝑇 → ((𝐻‘𝑥)𝑆(𝐻‘𝑦) ↔ (𝐻‘𝑥)𝑇(𝐻‘𝑦)))
2	1	bibi2d 342	. . . 4 ⊢ (𝑆 = 𝑇 → ((𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦)) ↔ (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑇(𝐻‘𝑦))))
3	2	2ralbidv 3215	. . 3 ⊢ (𝑆 = 𝑇 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦)) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑇(𝐻‘𝑦))))
4	3	anbi2d 629	. 2 ⊢ (𝑆 = 𝑇 → ((𝐻:𝐴–1-1-onto→𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦))) ↔ (𝐻:𝐴–1-1-onto→𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑇(𝐻‘𝑦)))))
5		df-isom 6557	. 2 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ (𝐻:𝐴–1-1-onto→𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦))))
6		df-isom 6557	. 2 ⊢ (𝐻 Isom 𝑅, 𝑇 (𝐴, 𝐵) ↔ (𝐻:𝐴–1-1-onto→𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑇(𝐻‘𝑦))))
7	4, 5, 6	3bitr4g 314	1 ⊢ (𝑆 = 𝑇 → (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐻 Isom 𝑅, 𝑇 (𝐴, 𝐵)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1534  ∀wral 3058   class class class wbr 5148  –1-1-onto→wf1o 6547  ‘cfv 6548   Isom wiso 6549

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699

This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1775  df-cleq 2720  df-clel 2806  df-ral 3059  df-br 5149  df-isom 6557

This theorem is referenced by:  fnwelem  8136  hartogslem1  9566  leiso  14453  gtiso  32493