Theorem rngimcnv 20365

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

Step	Hyp	Ref	Expression
1		rngimrcl 20355	. 2 ⊢ (𝐹 ∈ (𝑆 RngIso 𝑇) → (𝑆 ∈ V ∧ 𝑇 ∈ V))
2		isrngim 20354	. . 3 ⊢ ((𝑆 ∈ V ∧ 𝑇 ∈ V) → (𝐹 ∈ (𝑆 RngIso 𝑇) ↔ (𝐹 ∈ (𝑆 RngHom 𝑇) ∧ ◡𝐹 ∈ (𝑇 RngHom 𝑆))))
3		eqid 2729	. . . . . . . 8 ⊢ (Base‘𝑆) = (Base‘𝑆)
4		eqid 2729	. . . . . . . 8 ⊢ (Base‘𝑇) = (Base‘𝑇)
5	3, 4	rnghmf 20357	. . . . . . 7 ⊢ (𝐹 ∈ (𝑆 RngHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
6		frel 6693	. . . . . . . 8 ⊢ (𝐹:(Base‘𝑆)⟶(Base‘𝑇) → Rel 𝐹)
7		dfrel2 6162	. . . . . . . 8 ⊢ (Rel 𝐹 ↔ ◡◡𝐹 = 𝐹)
8	6, 7	sylib 218	. . . . . . 7 ⊢ (𝐹:(Base‘𝑆)⟶(Base‘𝑇) → ◡◡𝐹 = 𝐹)
9	5, 8	syl 17	. . . . . 6 ⊢ (𝐹 ∈ (𝑆 RngHom 𝑇) → ◡◡𝐹 = 𝐹)
10		id 22	. . . . . 6 ⊢ (𝐹 ∈ (𝑆 RngHom 𝑇) → 𝐹 ∈ (𝑆 RngHom 𝑇))
11	9, 10	eqeltrd 2828	. . . . 5 ⊢ (𝐹 ∈ (𝑆 RngHom 𝑇) → ◡◡𝐹 ∈ (𝑆 RngHom 𝑇))
12	11	anim1ci 616	. . . 4 ⊢ ((𝐹 ∈ (𝑆 RngHom 𝑇) ∧ ◡𝐹 ∈ (𝑇 RngHom 𝑆)) → (◡𝐹 ∈ (𝑇 RngHom 𝑆) ∧ ◡◡𝐹 ∈ (𝑆 RngHom 𝑇)))
13		isrngim 20354	. . . . 5 ⊢ ((𝑇 ∈ V ∧ 𝑆 ∈ V) → (◡𝐹 ∈ (𝑇 RngIso 𝑆) ↔ (◡𝐹 ∈ (𝑇 RngHom 𝑆) ∧ ◡◡𝐹 ∈ (𝑆 RngHom 𝑇))))
14	13	ancoms 458	. . . 4 ⊢ ((𝑆 ∈ V ∧ 𝑇 ∈ V) → (◡𝐹 ∈ (𝑇 RngIso 𝑆) ↔ (◡𝐹 ∈ (𝑇 RngHom 𝑆) ∧ ◡◡𝐹 ∈ (𝑆 RngHom 𝑇))))
15	12, 14	imbitrrid 246	. . 3 ⊢ ((𝑆 ∈ V ∧ 𝑇 ∈ V) → ((𝐹 ∈ (𝑆 RngHom 𝑇) ∧ ◡𝐹 ∈ (𝑇 RngHom 𝑆)) → ◡𝐹 ∈ (𝑇 RngIso 𝑆)))
16	2, 15	sylbid 240	. 2 ⊢ ((𝑆 ∈ V ∧ 𝑇 ∈ V) → (𝐹 ∈ (𝑆 RngIso 𝑇) → ◡𝐹 ∈ (𝑇 RngIso 𝑆)))
17	1, 16	mpcom 38	1 ⊢ (𝐹 ∈ (𝑆 RngIso 𝑇) → ◡𝐹 ∈ (𝑇 RngIso 𝑆))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3447  ^◡ccnv 5637  Rel wrel 5643  ⟶wf 6507  ‘cfv 6511  (class class class)co 7387  Basecbs 17179   RngHom crnghm 20343   RngIso crngim 20344

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-fv 6519  df-ov 7390  df-oprab 7391  df-mpo 7392  df-1st 7968  df-2nd 7969  df-map 8801  df-ghm 19145  df-abl 19713  df-rng 20062  df-rnghm 20345  df-rngim 20346

This theorem is referenced by:  rngisom1  20375  rngringbdlem2  21217  rngqiprngu  21228