Theorem risci 38027

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)) → 𝑅 ≃𝑟 𝑆)

Proof of Theorem risci

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex2 2808	. . 3 ⊢ (𝐹 ∈ (𝑅 RingOpsIso 𝑆) → ∃𝑓 𝑓 ∈ (𝑅 RingOpsIso 𝑆))
2		risc 38026	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) → (𝑅 ≃𝑟 𝑆 ↔ ∃𝑓 𝑓 ∈ (𝑅 RingOpsIso 𝑆)))
3	1, 2	imbitrrid 246	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) → (𝐹 ∈ (𝑅 RingOpsIso 𝑆) → 𝑅 ≃𝑟 𝑆))
4	3	3impia 1117	1 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsIso 𝑆)) → 𝑅 ≃𝑟 𝑆)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086  ∃wex 1780   ∈ wcel 2111   class class class wbr 5086  (class class class)co 7341  RingOpscrngo 37934   RingOpsIso crngoiso 38001   ≃_𝑟 crisc 38002

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-sep 5229  ax-nul 5239  ax-pr 5365

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4279  df-if 4471  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-br 5087  df-opab 5149  df-iota 6432  df-fv 6484  df-ov 7344  df-risc 38023

This theorem is referenced by:  riscer  38028