Theorem rngimcnv 20371

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 20361	. 2 ⊢ (𝐹 ∈ (𝑆 RngIso 𝑇) → (𝑆 ∈ V ∧ 𝑇 ∈ V))
2		isrngim 20360	. . 3 ⊢ ((𝑆 ∈ V ∧ 𝑇 ∈ V) → (𝐹 ∈ (𝑆 RngIso 𝑇) ↔ (𝐹 ∈ (𝑆 RngHom 𝑇) ∧ ◡𝐹 ∈ (𝑇 RngHom 𝑆))))
3		eqid 2730	. . . . . . . 8 ⊢ (Base‘𝑆) = (Base‘𝑆)
4		eqid 2730	. . . . . . . 8 ⊢ (Base‘𝑇) = (Base‘𝑇)
5	3, 4	rnghmf 20363	. . . . . . 7 ⊢ (𝐹 ∈ (𝑆 RngHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
6		frel 6695	. . . . . . . 8 ⊢ (𝐹:(Base‘𝑆)⟶(Base‘𝑇) → Rel 𝐹)
7		dfrel2 6164	. . . . . . . 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 2829	. . . . 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 20360	. . . . 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 3450  ^◡ccnv 5639  Rel wrel 5645  ⟶wf 6509  ‘cfv 6513  (class class class)co 7389  Basecbs 17185   RngHom crnghm 20349   RngIso crngim 20350

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 2702  ax-sep 5253  ax-nul 5263  ax-pow 5322  ax-pr 5389  ax-un 7713

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 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-sbc 3756  df-csb 3865  df-dif 3919  df-un 3921  df-in 3923  df-ss 3933  df-nul 4299  df-if 4491  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-iun 4959  df-br 5110  df-opab 5172  df-mpt 5191  df-id 5535  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-res 5652  df-ima 5653  df-iota 6466  df-fun 6515  df-fn 6516  df-f 6517  df-fv 6521  df-ov 7392  df-oprab 7393  df-mpo 7394  df-1st 7970  df-2nd 7971  df-map 8803  df-ghm 19151  df-abl 19719  df-rng 20068  df-rnghm 20351  df-rngim 20352

This theorem is referenced by:  rngisom1  20381  rngringbdlem2  21223  rngqiprngu  21234