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Theorem brric 20521
Description: The relation "is isomorphic to" for (unital) rings. (Contributed by AV, 24-Dec-2019.)
Assertion
Ref Expression
brric (𝑅𝑟 𝑆 ↔ (𝑅 RingIso 𝑆) ≠ ∅)

Proof of Theorem brric
Dummy variables 𝑟 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-ric 20492 . 2 𝑟 = ( RingIso “ (V ∖ 1o))
2 ovex 7464 . . . . 5 (𝑟 RingHom 𝑠) ∈ V
3 rabexg 5343 . . . . 5 ((𝑟 RingHom 𝑠) ∈ V → { ∈ (𝑟 RingHom 𝑠) ∣ ∈ (𝑠 RingHom 𝑟)} ∈ V)
42, 3mp1i 13 . . . 4 ((𝑟 ∈ V ∧ 𝑠 ∈ V) → { ∈ (𝑟 RingHom 𝑠) ∣ ∈ (𝑠 RingHom 𝑟)} ∈ V)
54rgen2 3197 . . 3 𝑟 ∈ V ∀𝑠 ∈ V { ∈ (𝑟 RingHom 𝑠) ∣ ∈ (𝑠 RingHom 𝑟)} ∈ V
6 df-rim 20490 . . . 4 RingIso = (𝑟 ∈ V, 𝑠 ∈ V ↦ { ∈ (𝑟 RingHom 𝑠) ∣ ∈ (𝑠 RingHom 𝑟)})
76fnmpo 8093 . . 3 (∀𝑟 ∈ V ∀𝑠 ∈ V { ∈ (𝑟 RingHom 𝑠) ∣ ∈ (𝑠 RingHom 𝑟)} ∈ V → RingIso Fn (V × V))
85, 7ax-mp 5 . 2 RingIso Fn (V × V)
91, 8brwitnlem 8544 1 (𝑅𝑟 𝑆 ↔ (𝑅 RingIso 𝑆) ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395  wcel 2106  wne 2938  wral 3059  {crab 3433  Vcvv 3478  c0 4339   class class class wbr 5148   × cxp 5687  ccnv 5688   Fn wfn 6558  (class class class)co 7431   RingHom crh 20486   RingIso crs 20487  𝑟 cric 20488
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-1o 8505  df-rim 20490  df-ric 20492
This theorem is referenced by:  brrici  20522  brric2  20523  mat1ric  22509  scmatric  22559  matcpmric  22781  pmmpric  22845  ricsym  42506  rictr  42507  riccrng1  42508  ricdrng1  42515
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