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Theorem isoeq2 7060
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 5059 . . . . 5 (𝑅 = 𝑇 → (𝑥𝑅𝑦𝑥𝑇𝑦))
21bibi1d 345 . . . 4 (𝑅 = 𝑇 → ((𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦)) ↔ (𝑥𝑇𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦))))
322ralbidv 3196 . . 3 (𝑅 = 𝑇 → (∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦)) ↔ ∀𝑥𝐴𝑦𝐴 (𝑥𝑇𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦))))
43anbi2d 628 . 2 (𝑅 = 𝑇 → ((𝐻:𝐴1-1-onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦))) ↔ (𝐻:𝐴1-1-onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑇𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦)))))
5 df-isom 6357 . 2 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ (𝐻:𝐴1-1-onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦))))
6 df-isom 6357 . 2 (𝐻 Isom 𝑇, 𝑆 (𝐴, 𝐵) ↔ (𝐻:𝐴1-1-onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑇𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦))))
74, 5, 63bitr4g 315 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 207  wa 396   = wceq 1528  wral 3135   class class class wbr 5057  1-1-ontowf1o 6347  cfv 6348   Isom wiso 6349
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1772  df-cleq 2811  df-clel 2890  df-ral 3140  df-br 5058  df-isom 6357
This theorem is referenced by:  leiso  13805  gtiso  30362  rrx2plordisom  44638
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