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Theorem rngimcnv 20456
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 20446 . 2 (𝐹 ∈ (𝑆 RngIso 𝑇) → (𝑆 ∈ V ∧ 𝑇 ∈ V))
2 isrngim 20445 . . 3 ((𝑆 ∈ V ∧ 𝑇 ∈ V) → (𝐹 ∈ (𝑆 RngIso 𝑇) ↔ (𝐹 ∈ (𝑆 RngHom 𝑇) ∧ 𝐹 ∈ (𝑇 RngHom 𝑆))))
3 eqid 2737 . . . . . . . 8 (Base‘𝑆) = (Base‘𝑆)
4 eqid 2737 . . . . . . . 8 (Base‘𝑇) = (Base‘𝑇)
53, 4rnghmf 20448 . . . . . . 7 (𝐹 ∈ (𝑆 RngHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
6 frel 6741 . . . . . . . 8 (𝐹:(Base‘𝑆)⟶(Base‘𝑇) → Rel 𝐹)
7 dfrel2 6209 . . . . . . . 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 2841 . . . . 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 20445 . . . . 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 1540  wcel 2108  Vcvv 3480  ccnv 5684  Rel wrel 5690  wf 6557  cfv 6561  (class class class)co 7431  Basecbs 17247   RngHom crnghm 20434   RngIso crngim 20435
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-map 8868  df-ghm 19231  df-abl 19801  df-rng 20150  df-rnghm 20436  df-rngim 20437
This theorem is referenced by:  rngisom1  20466  rngringbdlem2  21317  rngqiprngu  21328
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