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Theorem brric 20504
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 20475 . 2 𝑟 = ( RingIso “ (V ∖ 1o))
2 ovex 7464 . . . . 5 (𝑟 RingHom 𝑠) ∈ V
3 rabexg 5337 . . . . 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 3199 . . 3 𝑟 ∈ V ∀𝑠 ∈ V { ∈ (𝑟 RingHom 𝑠) ∣ ∈ (𝑠 RingHom 𝑟)} ∈ V
6 df-rim 20473 . . . 4 RingIso = (𝑟 ∈ V, 𝑠 ∈ V ↦ { ∈ (𝑟 RingHom 𝑠) ∣ ∈ (𝑠 RingHom 𝑟)})
76fnmpo 8094 . . 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 8545 1 (𝑅𝑟 𝑆 ↔ (𝑅 RingIso 𝑆) ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395  wcel 2108  wne 2940  wral 3061  {crab 3436  Vcvv 3480  c0 4333   class class class wbr 5143   × cxp 5683  ccnv 5684   Fn wfn 6556  (class class class)co 7431   RingHom crh 20469   RingIso crs 20470  𝑟 cric 20471
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-1o 8506  df-rim 20473  df-ric 20475
This theorem is referenced by:  brrici  20505  brric2  20506  mat1ric  22493  scmatric  22543  matcpmric  22765  pmmpric  22829  ricsym  42529  rictr  42530  riccrng1  42531  ricdrng1  42538
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