MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  isoeq3 Structured version   Visualization version   GIF version

Theorem isoeq3 7311
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 5143 . . . . 5 (𝑆 = 𝑇 → ((𝐻𝑥)𝑆(𝐻𝑦) ↔ (𝐻𝑥)𝑇(𝐻𝑦)))
21bibi2d 342 . . . 4 (𝑆 = 𝑇 → ((𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦)) ↔ (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑇(𝐻𝑦))))
322ralbidv 3212 . . 3 (𝑆 = 𝑇 → (∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦)) ↔ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑇(𝐻𝑦))))
43anbi2d 628 . 2 (𝑆 = 𝑇 → ((𝐻:𝐴1-1-onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦))) ↔ (𝐻:𝐴1-1-onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑇(𝐻𝑦)))))
5 df-isom 6545 . 2 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ (𝐻:𝐴1-1-onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦))))
6 df-isom 6545 . 2 (𝐻 Isom 𝑅, 𝑇 (𝐴, 𝐵) ↔ (𝐻:𝐴1-1-onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑇(𝐻𝑦))))
74, 5, 63bitr4g 314 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 205  wa 395   = wceq 1533  wral 3055   class class class wbr 5141  1-1-ontowf1o 6535  cfv 6536   Isom wiso 6537
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1774  df-cleq 2718  df-clel 2804  df-ral 3056  df-br 5142  df-isom 6545
This theorem is referenced by:  fnwelem  8114  hartogslem1  9536  leiso  14424  gtiso  32427
  Copyright terms: Public domain W3C validator