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Theorem isriscg 34088
 Description: The ring isomorphism relation. (Contributed by Jeff Madsen, 16-Jun-2011.)
Assertion
Ref Expression
isriscg ((𝑅𝐴𝑆𝐵) → (𝑅𝑟 𝑆 ↔ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ ∃𝑓 𝑓 ∈ (𝑅 RngIso 𝑆))))
Distinct variable groups:   𝑅,𝑓   𝑆,𝑓
Allowed substitution hints:   𝐴(𝑓)   𝐵(𝑓)

Proof of Theorem isriscg
Dummy variables 𝑟 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eleq1 2819 . . . 4 (𝑟 = 𝑅 → (𝑟 ∈ RingOps ↔ 𝑅 ∈ RingOps))
21anbi1d 743 . . 3 (𝑟 = 𝑅 → ((𝑟 ∈ RingOps ∧ 𝑠 ∈ RingOps) ↔ (𝑅 ∈ RingOps ∧ 𝑠 ∈ RingOps)))
3 oveq1 6812 . . . . 5 (𝑟 = 𝑅 → (𝑟 RngIso 𝑠) = (𝑅 RngIso 𝑠))
43eleq2d 2817 . . . 4 (𝑟 = 𝑅 → (𝑓 ∈ (𝑟 RngIso 𝑠) ↔ 𝑓 ∈ (𝑅 RngIso 𝑠)))
54exbidv 1991 . . 3 (𝑟 = 𝑅 → (∃𝑓 𝑓 ∈ (𝑟 RngIso 𝑠) ↔ ∃𝑓 𝑓 ∈ (𝑅 RngIso 𝑠)))
62, 5anbi12d 749 . 2 (𝑟 = 𝑅 → (((𝑟 ∈ RingOps ∧ 𝑠 ∈ RingOps) ∧ ∃𝑓 𝑓 ∈ (𝑟 RngIso 𝑠)) ↔ ((𝑅 ∈ RingOps ∧ 𝑠 ∈ RingOps) ∧ ∃𝑓 𝑓 ∈ (𝑅 RngIso 𝑠))))
7 eleq1 2819 . . . 4 (𝑠 = 𝑆 → (𝑠 ∈ RingOps ↔ 𝑆 ∈ RingOps))
87anbi2d 742 . . 3 (𝑠 = 𝑆 → ((𝑅 ∈ RingOps ∧ 𝑠 ∈ RingOps) ↔ (𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps)))
9 oveq2 6813 . . . . 5 (𝑠 = 𝑆 → (𝑅 RngIso 𝑠) = (𝑅 RngIso 𝑆))
109eleq2d 2817 . . . 4 (𝑠 = 𝑆 → (𝑓 ∈ (𝑅 RngIso 𝑠) ↔ 𝑓 ∈ (𝑅 RngIso 𝑆)))
1110exbidv 1991 . . 3 (𝑠 = 𝑆 → (∃𝑓 𝑓 ∈ (𝑅 RngIso 𝑠) ↔ ∃𝑓 𝑓 ∈ (𝑅 RngIso 𝑆)))
128, 11anbi12d 749 . 2 (𝑠 = 𝑆 → (((𝑅 ∈ RingOps ∧ 𝑠 ∈ RingOps) ∧ ∃𝑓 𝑓 ∈ (𝑅 RngIso 𝑠)) ↔ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ ∃𝑓 𝑓 ∈ (𝑅 RngIso 𝑆))))
13 df-risc 34087 . 2 𝑟 = {⟨𝑟, 𝑠⟩ ∣ ((𝑟 ∈ RingOps ∧ 𝑠 ∈ RingOps) ∧ ∃𝑓 𝑓 ∈ (𝑟 RngIso 𝑠))}
146, 12, 13brabg 5136 1 ((𝑅𝐴𝑆𝐵) → (𝑅𝑟 𝑆 ↔ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ ∃𝑓 𝑓 ∈ (𝑅 RngIso 𝑆))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   = wceq 1624  ∃wex 1845   ∈ wcel 2131   class class class wbr 4796  (class class class)co 6805  RingOpscrngo 33998   RngIso crngiso 34065   ≃𝑟 crisc 34066 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-sep 4925  ax-nul 4933  ax-pr 5047 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-rex 3048  df-rab 3051  df-v 3334  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-sn 4314  df-pr 4316  df-op 4320  df-uni 4581  df-br 4797  df-opab 4857  df-iota 6004  df-fv 6049  df-ov 6808  df-risc 34087 This theorem is referenced by:  isrisc  34089  risc  34090
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