Theorem rngimcnv 46690

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 RngIsom 𝑇) → ◡𝐹 ∈ (𝑇 RngIsom 𝑆))

Proof of Theorem rngimcnv

Step	Hyp	Ref	Expression
1		rngimrcl 46680	. 2 ⊢ (𝐹 ∈ (𝑆 RngIsom 𝑇) → (𝑆 ∈ V ∧ 𝑇 ∈ V))
2		isrngisom 46679	. . 3 ⊢ ((𝑆 ∈ V ∧ 𝑇 ∈ V) → (𝐹 ∈ (𝑆 RngIsom 𝑇) ↔ (𝐹 ∈ (𝑆 RngHomo 𝑇) ∧ ◡𝐹 ∈ (𝑇 RngHomo 𝑆))))
3		eqid 2732	. . . . . . . 8 ⊢ (Base‘𝑆) = (Base‘𝑆)
4		eqid 2732	. . . . . . . 8 ⊢ (Base‘𝑇) = (Base‘𝑇)
5	3, 4	rnghmf 46682	. . . . . . 7 ⊢ (𝐹 ∈ (𝑆 RngHomo 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
6		frel 6719	. . . . . . . 8 ⊢ (𝐹:(Base‘𝑆)⟶(Base‘𝑇) → Rel 𝐹)
7		dfrel2 6185	. . . . . . . 8 ⊢ (Rel 𝐹 ↔ ◡◡𝐹 = 𝐹)
8	6, 7	sylib 217	. . . . . . 7 ⊢ (𝐹:(Base‘𝑆)⟶(Base‘𝑇) → ◡◡𝐹 = 𝐹)
9	5, 8	syl 17	. . . . . 6 ⊢ (𝐹 ∈ (𝑆 RngHomo 𝑇) → ◡◡𝐹 = 𝐹)
10		id 22	. . . . . 6 ⊢ (𝐹 ∈ (𝑆 RngHomo 𝑇) → 𝐹 ∈ (𝑆 RngHomo 𝑇))
11	9, 10	eqeltrd 2833	. . . . 5 ⊢ (𝐹 ∈ (𝑆 RngHomo 𝑇) → ◡◡𝐹 ∈ (𝑆 RngHomo 𝑇))
12	11	anim1ci 616	. . . 4 ⊢ ((𝐹 ∈ (𝑆 RngHomo 𝑇) ∧ ◡𝐹 ∈ (𝑇 RngHomo 𝑆)) → (◡𝐹 ∈ (𝑇 RngHomo 𝑆) ∧ ◡◡𝐹 ∈ (𝑆 RngHomo 𝑇)))
13		isrngisom 46679	. . . . 5 ⊢ ((𝑇 ∈ V ∧ 𝑆 ∈ V) → (◡𝐹 ∈ (𝑇 RngIsom 𝑆) ↔ (◡𝐹 ∈ (𝑇 RngHomo 𝑆) ∧ ◡◡𝐹 ∈ (𝑆 RngHomo 𝑇))))
14	13	ancoms 459	. . . 4 ⊢ ((𝑆 ∈ V ∧ 𝑇 ∈ V) → (◡𝐹 ∈ (𝑇 RngIsom 𝑆) ↔ (◡𝐹 ∈ (𝑇 RngHomo 𝑆) ∧ ◡◡𝐹 ∈ (𝑆 RngHomo 𝑇))))
15	12, 14	imbitrrid 245	. . 3 ⊢ ((𝑆 ∈ V ∧ 𝑇 ∈ V) → ((𝐹 ∈ (𝑆 RngHomo 𝑇) ∧ ◡𝐹 ∈ (𝑇 RngHomo 𝑆)) → ◡𝐹 ∈ (𝑇 RngIsom 𝑆)))
16	2, 15	sylbid 239	. 2 ⊢ ((𝑆 ∈ V ∧ 𝑇 ∈ V) → (𝐹 ∈ (𝑆 RngIsom 𝑇) → ◡𝐹 ∈ (𝑇 RngIsom 𝑆)))
17	1, 16	mpcom 38	1 ⊢ (𝐹 ∈ (𝑆 RngIsom 𝑇) → ◡𝐹 ∈ (𝑇 RngIsom 𝑆))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  Vcvv 3474  ^◡ccnv 5674  Rel wrel 5680  ⟶wf 6536  ‘cfv 6540  (class class class)co 7405  Basecbs 17140   RngHomo crngh 46668   RngIsom crngs 46669

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7408  df-oprab 7409  df-mpo 7410  df-map 8818  df-ghm 19084  df-abl 19645  df-rng 46635  df-rnghomo 46670  df-rngisom 46671

This theorem is referenced by:  rngisom1  46703  rngringbdlem2  46772