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Theorem isrisc 36143
Description: The ring isomorphism relation. (Contributed by Jeff Madsen, 16-Jun-2011.)
Hypotheses
Ref Expression
isrisc.1 𝑅 ∈ V
isrisc.2 𝑆 ∈ V
Assertion
Ref Expression
isrisc (𝑅𝑟 𝑆 ↔ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ ∃𝑓 𝑓 ∈ (𝑅 RngIso 𝑆)))
Distinct variable groups:   𝑅,𝑓   𝑆,𝑓

Proof of Theorem isrisc
StepHypRef Expression
1 isrisc.1 . 2 𝑅 ∈ V
2 isrisc.2 . 2 𝑆 ∈ V
3 isriscg 36142 . 2 ((𝑅 ∈ V ∧ 𝑆 ∈ V) → (𝑅𝑟 𝑆 ↔ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ ∃𝑓 𝑓 ∈ (𝑅 RngIso 𝑆))))
41, 2, 3mp2an 689 1 (𝑅𝑟 𝑆 ↔ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ ∃𝑓 𝑓 ∈ (𝑅 RngIso 𝑆)))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 396  wex 1782  wcel 2106  Vcvv 3432   class class class wbr 5074  (class class class)co 7275  RingOpscrngo 36052   RngIso crngiso 36119  𝑟 crisc 36120
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-iota 6391  df-fv 6441  df-ov 7278  df-risc 36141
This theorem is referenced by:  riscer  36146
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