Theorem issubrng 20432

Description: The subring of non-unital ring predicate. (Contributed by AV, 14-Feb-2025.)

Hypothesis

Ref	Expression
issubrng.b	⊢ 𝐵 = (Base‘𝑅)

Assertion

Ref	Expression
issubrng	⊢ (𝐴 ∈ (SubRng‘𝑅) ↔ (𝑅 ∈ Rng ∧ (𝑅 ↾s 𝐴) ∈ Rng ∧ 𝐴 ⊆ 𝐵))

Proof of Theorem issubrng

Dummy variables 𝑤 𝑠 𝑟 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-subrng 20431	. . 3 ⊢ SubRng = (𝑤 ∈ Rng ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Rng})
2	1	mptrcl 6939	. 2 ⊢ (𝐴 ∈ (SubRng‘𝑅) → 𝑅 ∈ Rng)
3		simp1 1136	. 2 ⊢ ((𝑅 ∈ Rng ∧ (𝑅 ↾s 𝐴) ∈ Rng ∧ 𝐴 ⊆ 𝐵) → 𝑅 ∈ Rng)
4		fveq2 6822	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
5	4	pweqd 4568	. . . . . 6 ⊢ (𝑟 = 𝑅 → 𝒫 (Base‘𝑟) = 𝒫 (Base‘𝑅))
6		oveq1 7356	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (𝑟 ↾s 𝑠) = (𝑅 ↾s 𝑠))
7	6	eleq1d 2813	. . . . . 6 ⊢ (𝑟 = 𝑅 → ((𝑟 ↾s 𝑠) ∈ Rng ↔ (𝑅 ↾s 𝑠) ∈ Rng))
8	5, 7	rabeqbidv 3413	. . . . 5 ⊢ (𝑟 = 𝑅 → {𝑠 ∈ 𝒫 (Base‘𝑟) ∣ (𝑟 ↾s 𝑠) ∈ Rng} = {𝑠 ∈ 𝒫 (Base‘𝑅) ∣ (𝑅 ↾s 𝑠) ∈ Rng})
9		df-subrng 20431	. . . . 5 ⊢ SubRng = (𝑟 ∈ Rng ↦ {𝑠 ∈ 𝒫 (Base‘𝑟) ∣ (𝑟 ↾s 𝑠) ∈ Rng})
10		fvex 6835	. . . . . . 7 ⊢ (Base‘𝑅) ∈ V
11	10	pwex 5319	. . . . . 6 ⊢ 𝒫 (Base‘𝑅) ∈ V
12	11	rabex 5278	. . . . 5 ⊢ {𝑠 ∈ 𝒫 (Base‘𝑅) ∣ (𝑅 ↾s 𝑠) ∈ Rng} ∈ V
13	8, 9, 12	fvmpt 6930	. . . 4 ⊢ (𝑅 ∈ Rng → (SubRng‘𝑅) = {𝑠 ∈ 𝒫 (Base‘𝑅) ∣ (𝑅 ↾s 𝑠) ∈ Rng})
14	13	eleq2d 2814	. . 3 ⊢ (𝑅 ∈ Rng → (𝐴 ∈ (SubRng‘𝑅) ↔ 𝐴 ∈ {𝑠 ∈ 𝒫 (Base‘𝑅) ∣ (𝑅 ↾s 𝑠) ∈ Rng}))
15		oveq2 7357	. . . . . 6 ⊢ (𝑠 = 𝐴 → (𝑅 ↾s 𝑠) = (𝑅 ↾s 𝐴))
16	15	eleq1d 2813	. . . . 5 ⊢ (𝑠 = 𝐴 → ((𝑅 ↾s 𝑠) ∈ Rng ↔ (𝑅 ↾s 𝐴) ∈ Rng))
17	16	elrab 3648	. . . 4 ⊢ (𝐴 ∈ {𝑠 ∈ 𝒫 (Base‘𝑅) ∣ (𝑅 ↾s 𝑠) ∈ Rng} ↔ (𝐴 ∈ 𝒫 (Base‘𝑅) ∧ (𝑅 ↾s 𝐴) ∈ Rng))
18		issubrng.b	. . . . . . . . 9 ⊢ 𝐵 = (Base‘𝑅)
19	18	eqcomi 2738	. . . . . . . 8 ⊢ (Base‘𝑅) = 𝐵
20	19	sseq2i 3965	. . . . . . 7 ⊢ (𝐴 ⊆ (Base‘𝑅) ↔ 𝐴 ⊆ 𝐵)
21	20	anbi2i 623	. . . . . 6 ⊢ (((𝑅 ↾s 𝐴) ∈ Rng ∧ 𝐴 ⊆ (Base‘𝑅)) ↔ ((𝑅 ↾s 𝐴) ∈ Rng ∧ 𝐴 ⊆ 𝐵))
22		ibar 528	. . . . . 6 ⊢ (𝑅 ∈ Rng → (((𝑅 ↾s 𝐴) ∈ Rng ∧ 𝐴 ⊆ 𝐵) ↔ (𝑅 ∈ Rng ∧ ((𝑅 ↾s 𝐴) ∈ Rng ∧ 𝐴 ⊆ 𝐵))))
23	21, 22	bitrid 283	. . . . 5 ⊢ (𝑅 ∈ Rng → (((𝑅 ↾s 𝐴) ∈ Rng ∧ 𝐴 ⊆ (Base‘𝑅)) ↔ (𝑅 ∈ Rng ∧ ((𝑅 ↾s 𝐴) ∈ Rng ∧ 𝐴 ⊆ 𝐵))))
24	10	elpw2 5273	. . . . . 6 ⊢ (𝐴 ∈ 𝒫 (Base‘𝑅) ↔ 𝐴 ⊆ (Base‘𝑅))
25	24	anbi2ci 625	. . . . 5 ⊢ ((𝐴 ∈ 𝒫 (Base‘𝑅) ∧ (𝑅 ↾s 𝐴) ∈ Rng) ↔ ((𝑅 ↾s 𝐴) ∈ Rng ∧ 𝐴 ⊆ (Base‘𝑅)))
26		3anass 1094	. . . . 5 ⊢ ((𝑅 ∈ Rng ∧ (𝑅 ↾s 𝐴) ∈ Rng ∧ 𝐴 ⊆ 𝐵) ↔ (𝑅 ∈ Rng ∧ ((𝑅 ↾s 𝐴) ∈ Rng ∧ 𝐴 ⊆ 𝐵)))
27	23, 25, 26	3bitr4g 314	. . . 4 ⊢ (𝑅 ∈ Rng → ((𝐴 ∈ 𝒫 (Base‘𝑅) ∧ (𝑅 ↾s 𝐴) ∈ Rng) ↔ (𝑅 ∈ Rng ∧ (𝑅 ↾s 𝐴) ∈ Rng ∧ 𝐴 ⊆ 𝐵)))
28	17, 27	bitrid 283	. . 3 ⊢ (𝑅 ∈ Rng → (𝐴 ∈ {𝑠 ∈ 𝒫 (Base‘𝑅) ∣ (𝑅 ↾s 𝑠) ∈ Rng} ↔ (𝑅 ∈ Rng ∧ (𝑅 ↾s 𝐴) ∈ Rng ∧ 𝐴 ⊆ 𝐵)))
29	14, 28	bitrd 279	. 2 ⊢ (𝑅 ∈ Rng → (𝐴 ∈ (SubRng‘𝑅) ↔ (𝑅 ∈ Rng ∧ (𝑅 ↾s 𝐴) ∈ Rng ∧ 𝐴 ⊆ 𝐵)))
30	2, 3, 29	pm5.21nii 378	1 ⊢ (𝐴 ∈ (SubRng‘𝑅) ↔ (𝑅 ∈ Rng ∧ (𝑅 ↾s 𝐴) ∈ Rng ∧ 𝐴 ⊆ 𝐵))

Syntax hints:   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  {crab 3394   ⊆ wss 3903  𝒫 cpw 4551  ‘cfv 6482  (class class class)co 7349  Basecbs 17120   ↾_s cress 17141  Rngcrng 20037  SubRngcsubrng 20430

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 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371

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 3395  df-v 3438  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-br 5093  df-opab 5155  df-mpt 5174  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6438  df-fun 6484  df-fv 6490  df-ov 7352  df-subrng 20431

This theorem is referenced by:  subrngss  20433  subrngid  20434  subrngrng  20435  subrngrcl  20436  issubrng2  20443  subsubrng  20448  subrngpropd  20453  subrgsubrng  20463  rng2idlsubrng  21172