Theorem rngimcnv 20691

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 20681	. 2 ⊢ (𝐹 ∈ (𝑆 RngIso 𝑇) → (𝑆 ∈ V ∧ 𝑇 ∈ V))
2		isrngim 20680	. . 3 ⊢ ((𝑆 ∈ V ∧ 𝑇 ∈ V) → (𝐹 ∈ (𝑆 RngIso 𝑇) ↔ (𝐹 ∈ (𝑆 RngHom 𝑇) ∧ ◡𝐹 ∈ (𝑇 RngHom 𝑆))))
3		eqid 2760	. . . . . . . 8 ⊢ (Base‘𝑆) = (Base‘𝑆)
4		eqid 2760	. . . . . . . 8 ⊢ (Base‘𝑇) = (Base‘𝑇)
5	3, 4	rnghmf 20683	. . . . . . 7 ⊢ (𝐹 ∈ (𝑆 RngHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
6		frel 6715	. . . . . . . 8 ⊢ (𝐹:(Base‘𝑆)⟶(Base‘𝑇) → Rel 𝐹)
7		dfrel2 6180	. . . . . . . 8 ⊢ (Rel 𝐹 ↔ ◡◡𝐹 = 𝐹)
8	6, 7	sylib 221	. . . . . . 7 ⊢ (𝐹:(Base‘𝑆)⟶(Base‘𝑇) → ◡◡𝐹 = 𝐹)
9	5, 8	syl 18	. . . . . 6 ⊢ (𝐹 ∈ (𝑆 RngHom 𝑇) → ◡◡𝐹 = 𝐹)
10		id 23	. . . . . 6 ⊢ (𝐹 ∈ (𝑆 RngHom 𝑇) → 𝐹 ∈ (𝑆 RngHom 𝑇))
11	9, 10	eqeltrd 2860	. . . . 5 ⊢ (𝐹 ∈ (𝑆 RngHom 𝑇) → ◡◡𝐹 ∈ (𝑆 RngHom 𝑇))
12	11	anim1ci 628	. . . 4 ⊢ ((𝐹 ∈ (𝑆 RngHom 𝑇) ∧ ◡𝐹 ∈ (𝑇 RngHom 𝑆)) → (◡𝐹 ∈ (𝑇 RngHom 𝑆) ∧ ◡◡𝐹 ∈ (𝑆 RngHom 𝑇)))
13		isrngim 20680	. . . . 5 ⊢ ((𝑇 ∈ V ∧ 𝑆 ∈ V) → (◡𝐹 ∈ (𝑇 RngIso 𝑆) ↔ (◡𝐹 ∈ (𝑇 RngHom 𝑆) ∧ ◡◡𝐹 ∈ (𝑆 RngHom 𝑇))))
14	13	ancoms 464	. . . 4 ⊢ ((𝑆 ∈ V ∧ 𝑇 ∈ V) → (◡𝐹 ∈ (𝑇 RngIso 𝑆) ↔ (◡𝐹 ∈ (𝑇 RngHom 𝑆) ∧ ◡◡𝐹 ∈ (𝑆 RngHom 𝑇))))
15	12, 14	imbitrrid 249	. . 3 ⊢ ((𝑆 ∈ V ∧ 𝑇 ∈ V) → ((𝐹 ∈ (𝑆 RngHom 𝑇) ∧ ◡𝐹 ∈ (𝑇 RngHom 𝑆)) → ◡𝐹 ∈ (𝑇 RngIso 𝑆)))
16	2, 15	sylbid 243	. 2 ⊢ ((𝑆 ∈ V ∧ 𝑇 ∈ V) → (𝐹 ∈ (𝑆 RngIso 𝑇) → ◡𝐹 ∈ (𝑇 RngIso 𝑆)))
17	1, 16	mpcom 39	1 ⊢ (𝐹 ∈ (𝑆 RngIso 𝑇) → ◡𝐹 ∈ (𝑇 RngIso 𝑆))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144  Vcvv 3450  ^◡ccnv 5649  Rel wrel 5655  ⟶wf 6534  ‘cfv 6538  (class class class)co 7420  Basecbs 17392   RngHom crnghm 20669   RngIso crngim 20670

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-fv 6546  df-ov 7423  df-oprab 7424  df-mpo 7425  df-1st 8001  df-2nd 8002  df-map 8852  df-ghm 19433  df-abl 20002  df-rng 20380  df-rnghm 20671  df-rngim 20672

This theorem is used by:  rngisom1  20701  rngringbdlem2  21608  rngqiprngu  21619