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Theorem isriscg 37364
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 RingOpsIso 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 2815 . . . 4 (𝑟 = 𝑅 → (𝑟 ∈ RingOps ↔ 𝑅 ∈ RingOps))
21anbi1d 629 . . 3 (𝑟 = 𝑅 → ((𝑟 ∈ RingOps ∧ 𝑠 ∈ RingOps) ↔ (𝑅 ∈ RingOps ∧ 𝑠 ∈ RingOps)))
3 oveq1 7411 . . . . 5 (𝑟 = 𝑅 → (𝑟 RingOpsIso 𝑠) = (𝑅 RingOpsIso 𝑠))
43eleq2d 2813 . . . 4 (𝑟 = 𝑅 → (𝑓 ∈ (𝑟 RingOpsIso 𝑠) ↔ 𝑓 ∈ (𝑅 RingOpsIso 𝑠)))
54exbidv 1916 . . 3 (𝑟 = 𝑅 → (∃𝑓 𝑓 ∈ (𝑟 RingOpsIso 𝑠) ↔ ∃𝑓 𝑓 ∈ (𝑅 RingOpsIso 𝑠)))
62, 5anbi12d 630 . 2 (𝑟 = 𝑅 → (((𝑟 ∈ RingOps ∧ 𝑠 ∈ RingOps) ∧ ∃𝑓 𝑓 ∈ (𝑟 RingOpsIso 𝑠)) ↔ ((𝑅 ∈ RingOps ∧ 𝑠 ∈ RingOps) ∧ ∃𝑓 𝑓 ∈ (𝑅 RingOpsIso 𝑠))))
7 eleq1 2815 . . . 4 (𝑠 = 𝑆 → (𝑠 ∈ RingOps ↔ 𝑆 ∈ RingOps))
87anbi2d 628 . . 3 (𝑠 = 𝑆 → ((𝑅 ∈ RingOps ∧ 𝑠 ∈ RingOps) ↔ (𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps)))
9 oveq2 7412 . . . . 5 (𝑠 = 𝑆 → (𝑅 RingOpsIso 𝑠) = (𝑅 RingOpsIso 𝑆))
109eleq2d 2813 . . . 4 (𝑠 = 𝑆 → (𝑓 ∈ (𝑅 RingOpsIso 𝑠) ↔ 𝑓 ∈ (𝑅 RingOpsIso 𝑆)))
1110exbidv 1916 . . 3 (𝑠 = 𝑆 → (∃𝑓 𝑓 ∈ (𝑅 RingOpsIso 𝑠) ↔ ∃𝑓 𝑓 ∈ (𝑅 RingOpsIso 𝑆)))
128, 11anbi12d 630 . 2 (𝑠 = 𝑆 → (((𝑅 ∈ RingOps ∧ 𝑠 ∈ RingOps) ∧ ∃𝑓 𝑓 ∈ (𝑅 RingOpsIso 𝑠)) ↔ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ ∃𝑓 𝑓 ∈ (𝑅 RingOpsIso 𝑆))))
13 df-risc 37363 . 2 𝑟 = {⟨𝑟, 𝑠⟩ ∣ ((𝑟 ∈ RingOps ∧ 𝑠 ∈ RingOps) ∧ ∃𝑓 𝑓 ∈ (𝑟 RingOpsIso 𝑠))}
146, 12, 13brabg 5532 1 ((𝑅𝐴𝑆𝐵) → (𝑅𝑟 𝑆 ↔ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ ∃𝑓 𝑓 ∈ (𝑅 RingOpsIso 𝑆))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1533  wex 1773  wcel 2098   class class class wbr 5141  (class class class)co 7404  RingOpscrngo 37274   RingOpsIso crngoiso 37341  𝑟 crisc 37342
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-iota 6488  df-fv 6544  df-ov 7407  df-risc 37363
This theorem is referenced by:  isrisc  37365  risc  37366
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