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Theorem rngimrcl 20338
Description: Reverse closure for an isomorphism of non-unital rings. (Contributed by AV, 22-Feb-2020.)
Assertion
Ref Expression
rngimrcl (𝐹 ∈ (𝑅 RngIso 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-rngim 20329 . 2 RngIso = (𝑟 ∈ V, 𝑠 ∈ V ↦ {𝑓 ∈ (𝑟 RngHom 𝑠) ∣ 𝑓 ∈ (𝑠 RngHom 𝑟)})
21elmpocl 7651 1 (𝐹 ∈ (𝑅 RngIso 𝑆) → (𝑅 ∈ V ∧ 𝑆 ∈ V))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2105  {crab 3431  Vcvv 3473  ccnv 5676  (class class class)co 7412   RngHom crnghm 20326   RngIso crngim 20327
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  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 7415  df-oprab 7416  df-mpo 7417  df-rngim 20329
This theorem is referenced by:  rngimf1o  20346  rngimrnghm  20347  rngimcnv  20348
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