Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  rngimrcl Structured version   Visualization version   GIF version

Theorem rngimrcl 46695
Description: Reverse closure for an isomorphism of non-unital rings. (Contributed by AV, 22-Feb-2020.)
Assertion
Ref Expression
rngimrcl (𝐹 ∈ (𝑅 RngIsom 𝑆) → (𝑅 ∈ V ∧ 𝑆 ∈ V))

Proof of Theorem rngimrcl
Dummy variables 𝑓 𝑟 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-rngisom 46686 . 2 RngIsom = (𝑟 ∈ V, 𝑠 ∈ V ↦ {𝑓 ∈ (𝑟 RngHomo 𝑠) ∣ 𝑓 ∈ (𝑠 RngHomo 𝑟)})
21elmpocl 7648 1 (𝐹 ∈ (𝑅 RngIsom 𝑆) → (𝑅 ∈ V ∧ 𝑆 ∈ V))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  wcel 2107  {crab 3433  Vcvv 3475  ccnv 5676  (class class class)co 7409   RngHomo crngh 46683   RngIsom crngs 46684
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-xp 5683  df-dm 5687  df-iota 6496  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-rngisom 46686
This theorem is referenced by:  rngimf1o  46703  rngimrnghm  46704  rngimcnv  46705
  Copyright terms: Public domain W3C validator