MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rimrcl Structured version   Visualization version   GIF version

Theorem rimrcl 20380
Description: Reverse closure for an isomorphism of rings. (Contributed by AV, 22-Oct-2019.)
Assertion
Ref Expression
rimrcl (𝐹 ∈ (𝑅 RingIso 𝑆) → (𝑅 ∈ V ∧ 𝑆 ∈ V))

Proof of Theorem rimrcl
Dummy variables 𝑓 𝑟 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-rim 20371 . 2 RingIso = (𝑟 ∈ V, 𝑠 ∈ V ↦ {𝑓 ∈ (𝑟 RingHom 𝑠) ∣ 𝑓 ∈ (𝑠 RingHom 𝑟)})
21elmpocl 7642 1 (𝐹 ∈ (𝑅 RingIso 𝑆) → (𝑅 ∈ V ∧ 𝑆 ∈ V))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2098  {crab 3424  Vcvv 3466  ccnv 5666  (class class class)co 7402   RingHom crh 20367   RingIso crs 20368
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-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418
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-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-opab 5202  df-xp 5673  df-dm 5677  df-iota 6486  df-fv 6542  df-ov 7405  df-oprab 7406  df-mpo 7407  df-rim 20371
This theorem is referenced by:  isrim0  20381  rimisrngim  20396
  Copyright terms: Public domain W3C validator