Theorem isoeq2 7357

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

Assertion

Ref	Expression
isoeq2	⊢ (𝑅 = 𝑇 → (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐻 Isom 𝑇, 𝑆 (𝐴, 𝐵)))

Proof of Theorem isoeq2

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

Step	Hyp	Ref	Expression
1		breq 5110	. . . . 5 ⊢ (𝑅 = 𝑇 → (𝑥𝑅𝑦 ↔ 𝑥𝑇𝑦))
2	1	bibi1d 346	. . . 4 ⊢ (𝑅 = 𝑇 → ((𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦)) ↔ (𝑥𝑇𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦))))
3	2	2ralbidv 3230	. . 3 ⊢ (𝑅 = 𝑇 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦)) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑇𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦))))
4	3	anbi2d 642	. 2 ⊢ (𝑅 = 𝑇 → ((𝐻:𝐴⟶1-1onto𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦))) ↔ (𝐻:𝐴⟶1-1onto𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑇𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦)))))
5		df-isom 6575	. 2 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ (𝐻:𝐴⟶1-1onto𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦))))
6		df-isom 6575	. 2 ⊢ (𝐻 Isom 𝑇, 𝑆 (𝐴, 𝐵) ↔ (𝐻:𝐴⟶1-1onto𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑇𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦))))
7	4, 5, 6	3bitr4g 317	1 ⊢ (𝑅 = 𝑇 → (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐻 Isom 𝑇, 𝑆 (𝐴, 𝐵)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570  ∀wral 3080   class class class wbr 5108  ⟶1-1ontowf1o 6565  ‘cfv 6566   Isom wiso 6567

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-cleq 2755  df-clel 2839  df-ral 3081  df-br 5109  df-isom 6575

This theorem is used by:  leiso  14645  gtiso  33391  rrx2plordisom  50167