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Theorem brric 19487
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 19461 . 2 𝑟 = ( RingIso “ (V ∖ 1o))
2 ovex 7173 . . . . 5 (𝑟 RingHom 𝑠) ∈ V
3 rabexg 5217 . . . . 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-rngiso 19459 . . . 4 RingIso = (𝑟 ∈ V, 𝑠 ∈ V ↦ { ∈ (𝑟 RingHom 𝑠) ∣ ∈ (𝑠 RingHom 𝑟)})
76fnmpo 7752 . . 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 8117 1 (𝑅𝑟 𝑆 ↔ (𝑅 RingIso 𝑆) ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399  wcel 2115  wne 3013  wral 3132  {crab 3136  Vcvv 3479  c0 4274   class class class wbr 5049   × cxp 5536  ccnv 5537   Fn wfn 6333  (class class class)co 7140   RingHom crh 19455   RingIso crs 19456  𝑟 cric 19457
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5186  ax-nul 5193  ax-pow 5249  ax-pr 5313  ax-un 7446
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3014  df-ral 3137  df-rex 3138  df-rab 3141  df-v 3481  df-sbc 3758  df-csb 3866  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-nul 4275  df-if 4449  df-sn 4549  df-pr 4551  df-op 4555  df-uni 4822  df-iun 4904  df-br 5050  df-opab 5112  df-mpt 5130  df-id 5443  df-xp 5544  df-rel 5545  df-cnv 5546  df-co 5547  df-dm 5548  df-rn 5549  df-res 5550  df-ima 5551  df-suc 6180  df-iota 6297  df-fun 6340  df-fn 6341  df-f 6342  df-fv 6346  df-ov 7143  df-oprab 7144  df-mpo 7145  df-1st 7674  df-2nd 7675  df-1o 8087  df-rngiso 19459  df-ric 19461
This theorem is referenced by:  brric2  19488  mat1ric  21084  scmatric  21134  matcpmric  21355  pmmpric  21419
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