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Theorem brric 18684
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 18658 . 2 𝑟 = ( RingIso “ (V ∖ 1𝑜))
2 ovex 6643 . . . . 5 (𝑟 RingHom 𝑠) ∈ V
3 rabexg 4782 . . . . 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)
54rgen2a 2973 . . 3 𝑟 ∈ V ∀𝑠 ∈ V { ∈ (𝑟 RingHom 𝑠) ∣ ∈ (𝑠 RingHom 𝑟)} ∈ V
6 df-rngiso 18656 . . . 4 RingIso = (𝑟 ∈ V, 𝑠 ∈ V ↦ { ∈ (𝑟 RingHom 𝑠) ∣ ∈ (𝑠 RingHom 𝑟)})
76fnmpt2 7198 . . 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 7547 1 (𝑅𝑟 𝑆 ↔ (𝑅 RingIso 𝑆) ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384  wcel 1987  wne 2790  wral 2908  {crab 2912  Vcvv 3190  c0 3897   class class class wbr 4623   × cxp 5082  ccnv 5083   Fn wfn 5852  (class class class)co 6615   RingHom crh 18652   RingIso crs 18653  𝑟 cric 18654
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-fv 5865  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-1st 7128  df-2nd 7129  df-1o 7520  df-rngiso 18656  df-ric 18658
This theorem is referenced by:  brric2  18685  mat1ric  20233  scmatric  20283  matcpmric  20504  pmmpric  20568
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