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Theorem risci 37973
Description: Determine that two rings are isomorphic. (Contributed by Jeff Madsen, 16-Jun-2011.)
Assertion
Ref Expression
risci ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsIso 𝑆)) → 𝑅𝑟 𝑆)

Proof of Theorem risci
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 elex2 2815 . . 3 (𝐹 ∈ (𝑅 RingOpsIso 𝑆) → ∃𝑓 𝑓 ∈ (𝑅 RingOpsIso 𝑆))
2 risc 37972 . . 3 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) → (𝑅𝑟 𝑆 ↔ ∃𝑓 𝑓 ∈ (𝑅 RingOpsIso 𝑆)))
31, 2imbitrrid 246 . 2 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) → (𝐹 ∈ (𝑅 RingOpsIso 𝑆) → 𝑅𝑟 𝑆))
433impia 1116 1 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsIso 𝑆)) → 𝑅𝑟 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086  wex 1775  wcel 2105   class class class wbr 5147  (class class class)co 7430  RingOpscrngo 37880   RingOpsIso crngoiso 37947  𝑟 crisc 37948
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-iota 6515  df-fv 6570  df-ov 7433  df-risc 37969
This theorem is referenced by:  riscer  37974
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