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Theorem rngimcnv 20473
Description: The converse of an isomorphism of non-unital rings is an isomorphism of non-unital rings. (Contributed by AV, 27-Feb-2025.)
Assertion
Ref Expression
rngimcnv (𝐹 ∈ (𝑆 RngIso 𝑇) → 𝐹 ∈ (𝑇 RngIso 𝑆))

Proof of Theorem rngimcnv
StepHypRef Expression
1 rngimrcl 20463 . 2 (𝐹 ∈ (𝑆 RngIso 𝑇) → (𝑆 ∈ V ∧ 𝑇 ∈ V))
2 isrngim 20462 . . 3 ((𝑆 ∈ V ∧ 𝑇 ∈ V) → (𝐹 ∈ (𝑆 RngIso 𝑇) ↔ (𝐹 ∈ (𝑆 RngHom 𝑇) ∧ 𝐹 ∈ (𝑇 RngHom 𝑆))))
3 eqid 2735 . . . . . . . 8 (Base‘𝑆) = (Base‘𝑆)
4 eqid 2735 . . . . . . . 8 (Base‘𝑇) = (Base‘𝑇)
53, 4rnghmf 20465 . . . . . . 7 (𝐹 ∈ (𝑆 RngHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
6 frel 6742 . . . . . . . 8 (𝐹:(Base‘𝑆)⟶(Base‘𝑇) → Rel 𝐹)
7 dfrel2 6211 . . . . . . . 8 (Rel 𝐹𝐹 = 𝐹)
86, 7sylib 218 . . . . . . 7 (𝐹:(Base‘𝑆)⟶(Base‘𝑇) → 𝐹 = 𝐹)
95, 8syl 17 . . . . . 6 (𝐹 ∈ (𝑆 RngHom 𝑇) → 𝐹 = 𝐹)
10 id 22 . . . . . 6 (𝐹 ∈ (𝑆 RngHom 𝑇) → 𝐹 ∈ (𝑆 RngHom 𝑇))
119, 10eqeltrd 2839 . . . . 5 (𝐹 ∈ (𝑆 RngHom 𝑇) → 𝐹 ∈ (𝑆 RngHom 𝑇))
1211anim1ci 616 . . . 4 ((𝐹 ∈ (𝑆 RngHom 𝑇) ∧ 𝐹 ∈ (𝑇 RngHom 𝑆)) → (𝐹 ∈ (𝑇 RngHom 𝑆) ∧ 𝐹 ∈ (𝑆 RngHom 𝑇)))
13 isrngim 20462 . . . . 5 ((𝑇 ∈ V ∧ 𝑆 ∈ V) → (𝐹 ∈ (𝑇 RngIso 𝑆) ↔ (𝐹 ∈ (𝑇 RngHom 𝑆) ∧ 𝐹 ∈ (𝑆 RngHom 𝑇))))
1413ancoms 458 . . . 4 ((𝑆 ∈ V ∧ 𝑇 ∈ V) → (𝐹 ∈ (𝑇 RngIso 𝑆) ↔ (𝐹 ∈ (𝑇 RngHom 𝑆) ∧ 𝐹 ∈ (𝑆 RngHom 𝑇))))
1512, 14imbitrrid 246 . . 3 ((𝑆 ∈ V ∧ 𝑇 ∈ V) → ((𝐹 ∈ (𝑆 RngHom 𝑇) ∧ 𝐹 ∈ (𝑇 RngHom 𝑆)) → 𝐹 ∈ (𝑇 RngIso 𝑆)))
162, 15sylbid 240 . 2 ((𝑆 ∈ V ∧ 𝑇 ∈ V) → (𝐹 ∈ (𝑆 RngIso 𝑇) → 𝐹 ∈ (𝑇 RngIso 𝑆)))
171, 16mpcom 38 1 (𝐹 ∈ (𝑆 RngIso 𝑇) → 𝐹 ∈ (𝑇 RngIso 𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  Vcvv 3478  ccnv 5688  Rel wrel 5694  wf 6559  cfv 6563  (class class class)co 7431  Basecbs 17245   RngHom crnghm 20451   RngIso crngim 20452
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-map 8867  df-ghm 19244  df-abl 19816  df-rng 20171  df-rnghm 20453  df-rngim 20454
This theorem is referenced by:  rngisom1  20483  rngringbdlem2  21335  rngqiprngu  21346
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