Theorem risci 38354

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 2816	. . 3 ⊢ (𝐹 ∈ (𝑅 RingOpsIso 𝑆) → ∃𝑓 𝑓 ∈ (𝑅 RingOpsIso 𝑆))
2		risc 38353	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) → (𝑅 ≃𝑟 𝑆 ↔ ∃𝑓 𝑓 ∈ (𝑅 RingOpsIso 𝑆)))
3	1, 2	imbitrrid 247	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) → (𝐹 ∈ (𝑅 RingOpsIso 𝑆) → 𝑅 ≃𝑟 𝑆))
4	3	3impia 1123	1 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsIso 𝑆)) → 𝑅 ≃𝑟 𝑆)

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1092  ∃wex 1786   ∈ wcel 2119   class class class wbr 5072  (class class class)co 7356  RingOpscrngo 38261   RingOpsIso crngoiso 38328   ≃_𝑟 crisc 38329

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711  ax-sep 5218  ax-pr 5362

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-opab 5135  df-iota 6441  df-fv 6493  df-ov 7359  df-risc 38350

This theorem is referenced by:  riscer  38355