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Theorem isoeq3 6797
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.
StepHypRef Expression
1 breq 4845 . . . . 5 (𝑆 = 𝑇 → ((𝐻𝑥)𝑆(𝐻𝑦) ↔ (𝐻𝑥)𝑇(𝐻𝑦)))
21bibi2d 334 . . . 4 (𝑆 = 𝑇 → ((𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦)) ↔ (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑇(𝐻𝑦))))
322ralbidv 3170 . . 3 (𝑆 = 𝑇 → (∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦)) ↔ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑇(𝐻𝑦))))
43anbi2d 623 . 2 (𝑆 = 𝑇 → ((𝐻:𝐴1-1-onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦))) ↔ (𝐻:𝐴1-1-onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑇(𝐻𝑦)))))
5 df-isom 6110 . 2 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ (𝐻:𝐴1-1-onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦))))
6 df-isom 6110 . 2 (𝐻 Isom 𝑅, 𝑇 (𝐴, 𝐵) ↔ (𝐻:𝐴1-1-onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑇(𝐻𝑦))))
74, 5, 63bitr4g 306 1 (𝑆 = 𝑇 → (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐻 Isom 𝑅, 𝑇 (𝐴, 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385   = wceq 1653  wral 3089   class class class wbr 4843  1-1-ontowf1o 6100  cfv 6101   Isom wiso 6102
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-ext 2777
This theorem depends on definitions:  df-bi 199  df-an 386  df-ex 1876  df-cleq 2792  df-clel 2795  df-ral 3094  df-br 4844  df-isom 6110
This theorem is referenced by:  fnwelem  7529  hartogslem1  8689  leiso  13492  gtiso  29996
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