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Theorem risci 32956
 Description: Determine that two rings are isomorphic. (Contributed by Jeff Madsen, 16-Jun-2011.)
Assertion
Ref Expression
risci ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) → 𝑅𝑟 𝑆)

Proof of Theorem risci
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 elex2 3189 . . 3 (𝐹 ∈ (𝑅 RngIso 𝑆) → ∃𝑓 𝑓 ∈ (𝑅 RngIso 𝑆))
2 risc 32955 . . 3 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) → (𝑅𝑟 𝑆 ↔ ∃𝑓 𝑓 ∈ (𝑅 RngIso 𝑆)))
31, 2syl5ibr 235 . 2 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) → (𝐹 ∈ (𝑅 RngIso 𝑆) → 𝑅𝑟 𝑆))
433impia 1253 1 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) → 𝑅𝑟 𝑆)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031  ∃wex 1695   ∈ wcel 1977   class class class wbr 4583  (class class class)co 6549  RingOpscrngo 32863   RngIso crngiso 32930   ≃𝑟 crisc 32931 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-iota 5768  df-fv 5812  df-ov 6552  df-risc 32952 This theorem is referenced by:  riscer  32957
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