Theorem issubrng 14480

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 14479	. . 3 ⊢ SubRng = (𝑤 ∈ Rng ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Rng})
2	1	mptrcl 5782	. 2 ⊢ (𝐴 ∈ (SubRng‘𝑅) → 𝑅 ∈ Rng)
3		simp1 1028	. 2 ⊢ ((𝑅 ∈ Rng ∧ (𝑅 ↾s 𝐴) ∈ Rng ∧ 𝐴 ⊆ 𝐵) → 𝑅 ∈ Rng)
4		df-subrng 14479	. . . . 5 ⊢ SubRng = (𝑟 ∈ Rng ↦ {𝑠 ∈ 𝒫 (Base‘𝑟) ∣ (𝑟 ↾s 𝑠) ∈ Rng})
5		fveq2 5690	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
6	5	pweqd 3690	. . . . . 6 ⊢ (𝑟 = 𝑅 → 𝒫 (Base‘𝑟) = 𝒫 (Base‘𝑅))
7		oveq1 6082	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (𝑟 ↾s 𝑠) = (𝑅 ↾s 𝑠))
8	7	eleq1d 2307	. . . . . 6 ⊢ (𝑟 = 𝑅 → ((𝑟 ↾s 𝑠) ∈ Rng ↔ (𝑅 ↾s 𝑠) ∈ Rng))
9	6, 8	rabeqbidv 2816	. . . . 5 ⊢ (𝑟 = 𝑅 → {𝑠 ∈ 𝒫 (Base‘𝑟) ∣ (𝑟 ↾s 𝑠) ∈ Rng} = {𝑠 ∈ 𝒫 (Base‘𝑅) ∣ (𝑅 ↾s 𝑠) ∈ Rng})
10		id 19	. . . . 5 ⊢ (𝑅 ∈ Rng → 𝑅 ∈ Rng)
11		basfn 13389	. . . . . . . 8 ⊢ Base Fn V
12		elex 2833	. . . . . . . 8 ⊢ (𝑅 ∈ Rng → 𝑅 ∈ V)
13		funfvex 5707	. . . . . . . . 9 ⊢ ((Fun Base ∧ 𝑅 ∈ dom Base) → (Base‘𝑅) ∈ V)
14	13	funfni 5478	. . . . . . . 8 ⊢ ((Base Fn V ∧ 𝑅 ∈ V) → (Base‘𝑅) ∈ V)
15	11, 12, 14	sylancr 418	. . . . . . 7 ⊢ (𝑅 ∈ Rng → (Base‘𝑅) ∈ V)
16	15	pwexd 4313	. . . . . 6 ⊢ (𝑅 ∈ Rng → 𝒫 (Base‘𝑅) ∈ V)
17		rabexg 4274	. . . . . 6 ⊢ (𝒫 (Base‘𝑅) ∈ V → {𝑠 ∈ 𝒫 (Base‘𝑅) ∣ (𝑅 ↾s 𝑠) ∈ Rng} ∈ V)
18	16, 17	syl 14	. . . . 5 ⊢ (𝑅 ∈ Rng → {𝑠 ∈ 𝒫 (Base‘𝑅) ∣ (𝑅 ↾s 𝑠) ∈ Rng} ∈ V)
19	4, 9, 10, 18	fvmptd3 5793	. . . 4 ⊢ (𝑅 ∈ Rng → (SubRng‘𝑅) = {𝑠 ∈ 𝒫 (Base‘𝑅) ∣ (𝑅 ↾s 𝑠) ∈ Rng})
20	19	eleq2d 2308	. . 3 ⊢ (𝑅 ∈ Rng → (𝐴 ∈ (SubRng‘𝑅) ↔ 𝐴 ∈ {𝑠 ∈ 𝒫 (Base‘𝑅) ∣ (𝑅 ↾s 𝑠) ∈ Rng}))
21		oveq2 6083	. . . . . 6 ⊢ (𝑠 = 𝐴 → (𝑅 ↾s 𝑠) = (𝑅 ↾s 𝐴))
22	21	eleq1d 2307	. . . . 5 ⊢ (𝑠 = 𝐴 → ((𝑅 ↾s 𝑠) ∈ Rng ↔ (𝑅 ↾s 𝐴) ∈ Rng))
23	22	elrab 2982	. . . 4 ⊢ (𝐴 ∈ {𝑠 ∈ 𝒫 (Base‘𝑅) ∣ (𝑅 ↾s 𝑠) ∈ Rng} ↔ (𝐴 ∈ 𝒫 (Base‘𝑅) ∧ (𝑅 ↾s 𝐴) ∈ Rng))
24		issubrng.b	. . . . . . . . 9 ⊢ 𝐵 = (Base‘𝑅)
25	24	eqcomi 2242	. . . . . . . 8 ⊢ (Base‘𝑅) = 𝐵
26	25	sseq2i 3275	. . . . . . 7 ⊢ (𝐴 ⊆ (Base‘𝑅) ↔ 𝐴 ⊆ 𝐵)
27	26	anbi2i 461	. . . . . 6 ⊢ (((𝑅 ↾s 𝐴) ∈ Rng ∧ 𝐴 ⊆ (Base‘𝑅)) ↔ ((𝑅 ↾s 𝐴) ∈ Rng ∧ 𝐴 ⊆ 𝐵))
28		ibar 301	. . . . . 6 ⊢ (𝑅 ∈ Rng → (((𝑅 ↾s 𝐴) ∈ Rng ∧ 𝐴 ⊆ 𝐵) ↔ (𝑅 ∈ Rng ∧ ((𝑅 ↾s 𝐴) ∈ Rng ∧ 𝐴 ⊆ 𝐵))))
29	27, 28	bitrid 192	. . . . 5 ⊢ (𝑅 ∈ Rng → (((𝑅 ↾s 𝐴) ∈ Rng ∧ 𝐴 ⊆ (Base‘𝑅)) ↔ (𝑅 ∈ Rng ∧ ((𝑅 ↾s 𝐴) ∈ Rng ∧ 𝐴 ⊆ 𝐵))))
30		ancom 266	. . . . . 6 ⊢ ((𝐴 ∈ 𝒫 (Base‘𝑅) ∧ (𝑅 ↾s 𝐴) ∈ Rng) ↔ ((𝑅 ↾s 𝐴) ∈ Rng ∧ 𝐴 ∈ 𝒫 (Base‘𝑅)))
31		elpw2g 4287	. . . . . . . 8 ⊢ ((Base‘𝑅) ∈ V → (𝐴 ∈ 𝒫 (Base‘𝑅) ↔ 𝐴 ⊆ (Base‘𝑅)))
32	15, 31	syl 14	. . . . . . 7 ⊢ (𝑅 ∈ Rng → (𝐴 ∈ 𝒫 (Base‘𝑅) ↔ 𝐴 ⊆ (Base‘𝑅)))
33	32	anbi2d 468	. . . . . 6 ⊢ (𝑅 ∈ Rng → (((𝑅 ↾s 𝐴) ∈ Rng ∧ 𝐴 ∈ 𝒫 (Base‘𝑅)) ↔ ((𝑅 ↾s 𝐴) ∈ Rng ∧ 𝐴 ⊆ (Base‘𝑅))))
34	30, 33	bitrid 192	. . . . 5 ⊢ (𝑅 ∈ Rng → ((𝐴 ∈ 𝒫 (Base‘𝑅) ∧ (𝑅 ↾s 𝐴) ∈ Rng) ↔ ((𝑅 ↾s 𝐴) ∈ Rng ∧ 𝐴 ⊆ (Base‘𝑅))))
35		3anass 1013	. . . . . 6 ⊢ ((𝑅 ∈ Rng ∧ (𝑅 ↾s 𝐴) ∈ Rng ∧ 𝐴 ⊆ 𝐵) ↔ (𝑅 ∈ Rng ∧ ((𝑅 ↾s 𝐴) ∈ Rng ∧ 𝐴 ⊆ 𝐵)))
36	35	a1i 9	. . . . 5 ⊢ (𝑅 ∈ Rng → ((𝑅 ∈ Rng ∧ (𝑅 ↾s 𝐴) ∈ Rng ∧ 𝐴 ⊆ 𝐵) ↔ (𝑅 ∈ Rng ∧ ((𝑅 ↾s 𝐴) ∈ Rng ∧ 𝐴 ⊆ 𝐵))))
37	29, 34, 36	3bitr4d 220	. . . 4 ⊢ (𝑅 ∈ Rng → ((𝐴 ∈ 𝒫 (Base‘𝑅) ∧ (𝑅 ↾s 𝐴) ∈ Rng) ↔ (𝑅 ∈ Rng ∧ (𝑅 ↾s 𝐴) ∈ Rng ∧ 𝐴 ⊆ 𝐵)))
38	23, 37	bitrid 192	. . 3 ⊢ (𝑅 ∈ Rng → (𝐴 ∈ {𝑠 ∈ 𝒫 (Base‘𝑅) ∣ (𝑅 ↾s 𝑠) ∈ Rng} ↔ (𝑅 ∈ Rng ∧ (𝑅 ↾s 𝐴) ∈ Rng ∧ 𝐴 ⊆ 𝐵)))
39	20, 38	bitrd 188	. 2 ⊢ (𝑅 ∈ Rng → (𝐴 ∈ (SubRng‘𝑅) ↔ (𝑅 ∈ Rng ∧ (𝑅 ↾s 𝐴) ∈ Rng ∧ 𝐴 ⊆ 𝐵)))
40	2, 3, 39	pm5.21nii 716	1 ⊢ (𝐴 ∈ (SubRng‘𝑅) ↔ (𝑅 ∈ Rng ∧ (𝑅 ↾s 𝐴) ∈ Rng ∧ 𝐴 ⊆ 𝐵))

Syntax hints:   ∧ wa 104   ↔ wb 105   ∧ w3a 1009   = wceq 1402   ∈ wcel 2209  {crab 2532  Vcvv 2821   ⊆ wss 3220  𝒫 cpw 3685   Fn wfn 5367  ‘cfv 5372  (class class class)co 6075  Basecbs 13330   ↾_s cress 13331  Rngcrng 14206  SubRngcsubrng 14478

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-cnex 8260  ax-resscn 8261  ax-1re 8263  ax-addrcl 8266

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-fv 5380  df-ov 6078  df-inn 9284  df-ndx 13333  df-slot 13334  df-base 13336  df-subrng 14479

This theorem is referenced by:  subrngss  14481  subrngid  14482  subrngrng  14483  subrngrcl  14484  issubrng2  14491  subsubrng  14495  subrngpropd  14497  rng2idlsubrng  14826