Theorem issubrng 14076

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 14075	. . 3 ⊢ SubRng = (𝑤 ∈ Rng ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Rng})
2	1	mptrcl 5685	. 2 ⊢ (𝐴 ∈ (SubRng‘𝑅) → 𝑅 ∈ Rng)
3		simp1 1000	. 2 ⊢ ((𝑅 ∈ Rng ∧ (𝑅 ↾s 𝐴) ∈ Rng ∧ 𝐴 ⊆ 𝐵) → 𝑅 ∈ Rng)
4		df-subrng 14075	. . . . 5 ⊢ SubRng = (𝑟 ∈ Rng ↦ {𝑠 ∈ 𝒫 (Base‘𝑟) ∣ (𝑟 ↾s 𝑠) ∈ Rng})
5		fveq2 5599	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
6	5	pweqd 3631	. . . . . 6 ⊢ (𝑟 = 𝑅 → 𝒫 (Base‘𝑟) = 𝒫 (Base‘𝑅))
7		oveq1 5974	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (𝑟 ↾s 𝑠) = (𝑅 ↾s 𝑠))
8	7	eleq1d 2276	. . . . . 6 ⊢ (𝑟 = 𝑅 → ((𝑟 ↾s 𝑠) ∈ Rng ↔ (𝑅 ↾s 𝑠) ∈ Rng))
9	6, 8	rabeqbidv 2771	. . . . 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 13005	. . . . . . . 8 ⊢ Base Fn V
12		elex 2788	. . . . . . . 8 ⊢ (𝑅 ∈ Rng → 𝑅 ∈ V)
13		funfvex 5616	. . . . . . . . 9 ⊢ ((Fun Base ∧ 𝑅 ∈ dom Base) → (Base‘𝑅) ∈ V)
14	13	funfni 5395	. . . . . . . 8 ⊢ ((Base Fn V ∧ 𝑅 ∈ V) → (Base‘𝑅) ∈ V)
15	11, 12, 14	sylancr 414	. . . . . . 7 ⊢ (𝑅 ∈ Rng → (Base‘𝑅) ∈ V)
16	15	pwexd 4241	. . . . . 6 ⊢ (𝑅 ∈ Rng → 𝒫 (Base‘𝑅) ∈ V)
17		rabexg 4203	. . . . . 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 5696	. . . 4 ⊢ (𝑅 ∈ Rng → (SubRng‘𝑅) = {𝑠 ∈ 𝒫 (Base‘𝑅) ∣ (𝑅 ↾s 𝑠) ∈ Rng})
20	19	eleq2d 2277	. . 3 ⊢ (𝑅 ∈ Rng → (𝐴 ∈ (SubRng‘𝑅) ↔ 𝐴 ∈ {𝑠 ∈ 𝒫 (Base‘𝑅) ∣ (𝑅 ↾s 𝑠) ∈ Rng}))
21		oveq2 5975	. . . . . 6 ⊢ (𝑠 = 𝐴 → (𝑅 ↾s 𝑠) = (𝑅 ↾s 𝐴))
22	21	eleq1d 2276	. . . . 5 ⊢ (𝑠 = 𝐴 → ((𝑅 ↾s 𝑠) ∈ Rng ↔ (𝑅 ↾s 𝐴) ∈ Rng))
23	22	elrab 2936	. . . 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 2211	. . . . . . . 8 ⊢ (Base‘𝑅) = 𝐵
26	25	sseq2i 3228	. . . . . . 7 ⊢ (𝐴 ⊆ (Base‘𝑅) ↔ 𝐴 ⊆ 𝐵)
27	26	anbi2i 457	. . . . . 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 4216	. . . . . . . 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 464	. . . . . 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 985	. . . . . 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 706	1 ⊢ (𝐴 ∈ (SubRng‘𝑅) ↔ (𝑅 ∈ Rng ∧ (𝑅 ↾s 𝐴) ∈ Rng ∧ 𝐴 ⊆ 𝐵))

Syntax hints:   ∧ wa 104   ↔ wb 105   ∧ w3a 981   = wceq 1373   ∈ wcel 2178  {crab 2490  Vcvv 2776   ⊆ wss 3174  𝒫 cpw 3626   Fn wfn 5285  ‘cfv 5290  (class class class)co 5967  Basecbs 12947   ↾_s cress 12948  Rngcrng 13809  SubRngcsubrng 14074

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-cnex 8051  ax-resscn 8052  ax-1re 8054  ax-addrcl 8057

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-fv 5298  df-ov 5970  df-inn 9072  df-ndx 12950  df-slot 12951  df-base 12953  df-subrng 14075

This theorem is referenced by:  subrngss  14077  subrngid  14078  subrngrng  14079  subrngrcl  14080  issubrng2  14087  subsubrng  14091  subrngpropd  14093  rng2idlsubrng  14394