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Theorem isoeq2 7055
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.
StepHypRef Expression
1 breq 5044 . . . . 5 (𝑅 = 𝑇 → (𝑥𝑅𝑦𝑥𝑇𝑦))
21bibi1d 347 . . . 4 (𝑅 = 𝑇 → ((𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦)) ↔ (𝑥𝑇𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦))))
322ralbidv 3189 . . 3 (𝑅 = 𝑇 → (∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦)) ↔ ∀𝑥𝐴𝑦𝐴 (𝑥𝑇𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦))))
43anbi2d 631 . 2 (𝑅 = 𝑇 → ((𝐻:𝐴1-1-onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦))) ↔ (𝐻:𝐴1-1-onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑇𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦)))))
5 df-isom 6343 . 2 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ (𝐻:𝐴1-1-onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦))))
6 df-isom 6343 . 2 (𝐻 Isom 𝑇, 𝑆 (𝐴, 𝐵) ↔ (𝐻:𝐴1-1-onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑇𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦))))
74, 5, 63bitr4g 317 1 (𝑅 = 𝑇 → (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐻 Isom 𝑇, 𝑆 (𝐴, 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wral 3130   class class class wbr 5042  1-1-ontowf1o 6333  cfv 6334   Isom wiso 6335
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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-ext 2794
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-cleq 2815  df-clel 2894  df-ral 3135  df-br 5043  df-isom 6343
This theorem is referenced by:  leiso  13813  gtiso  30444  rrx2plordisom  45076
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