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Theorem rngimrcl 44514
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 44505 . 2 RngIsom = (𝑟 ∈ V, 𝑠 ∈ V ↦ {𝑓 ∈ (𝑟 RngHomo 𝑠) ∣ 𝑓 ∈ (𝑠 RngHomo 𝑟)})
21elmpocl 7371 1 (𝐹 ∈ (𝑅 RngIsom 𝑆) → (𝑅 ∈ V ∧ 𝑆 ∈ V))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  wcel 2112  {crab 3113  Vcvv 3444  ccnv 5522  (class class class)co 7139   RngHomo crngh 44502   RngIsom crngs 44503
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298
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 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-v 3446  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-xp 5529  df-dm 5533  df-iota 6287  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144  df-rngisom 44505
This theorem is referenced by:  rngimf1o  44522  rngimrnghm  44523
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