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Theorem isrisc 36447
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 36446 . 2 ((𝑅 ∈ V ∧ 𝑆 ∈ V) → (𝑅𝑟 𝑆 ↔ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ ∃𝑓 𝑓 ∈ (𝑅 RngIso 𝑆))))
41, 2, 3mp2an 691 1 (𝑅𝑟 𝑆 ↔ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ ∃𝑓 𝑓 ∈ (𝑅 RngIso 𝑆)))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 397  wex 1782  wcel 2107  Vcvv 3446   class class class wbr 5106  (class class class)co 7358  RingOpscrngo 36356   RngIso crngiso 36423  𝑟 crisc 36424
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-rab 3409  df-v 3448  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-iota 6449  df-fv 6505  df-ov 7361  df-risc 36445
This theorem is referenced by:  riscer  36450
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