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Theorem isrisc 35416
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 35415 . 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 209  wa 399  wex 1781  wcel 2112  Vcvv 3444   class class class wbr 5033  (class class class)co 7139  RingOpscrngo 35325   RngIso crngiso 35392  𝑟 crisc 35393
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-v 3446  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-iota 6287  df-fv 6336  df-ov 7142  df-risc 35414
This theorem is referenced by:  riscer  35419
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