Theorem issubrng 14036

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 14035	. . 3 ⊢ SubRng = (𝑤 ∈ Rng ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Rng})
2	1	mptrcl 5675	. 2 ⊢ (𝐴 ∈ (SubRng‘𝑅) → 𝑅 ∈ Rng)
3		simp1 1000	. 2 ⊢ ((𝑅 ∈ Rng ∧ (𝑅 ↾s 𝐴) ∈ Rng ∧ 𝐴 ⊆ 𝐵) → 𝑅 ∈ Rng)
4		df-subrng 14035	. . . . 5 ⊢ SubRng = (𝑟 ∈ Rng ↦ {𝑠 ∈ 𝒫 (Base‘𝑟) ∣ (𝑟 ↾s 𝑠) ∈ Rng})
5		fveq2 5589	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
6	5	pweqd 3626	. . . . . 6 ⊢ (𝑟 = 𝑅 → 𝒫 (Base‘𝑟) = 𝒫 (Base‘𝑅))
7		oveq1 5964	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (𝑟 ↾s 𝑠) = (𝑅 ↾s 𝑠))
8	7	eleq1d 2275	. . . . . 6 ⊢ (𝑟 = 𝑅 → ((𝑟 ↾s 𝑠) ∈ Rng ↔ (𝑅 ↾s 𝑠) ∈ Rng))
9	6, 8	rabeqbidv 2768	. . . . 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 12965	. . . . . . . 8 ⊢ Base Fn V
12		elex 2785	. . . . . . . 8 ⊢ (𝑅 ∈ Rng → 𝑅 ∈ V)
13		funfvex 5606	. . . . . . . . 9 ⊢ ((Fun Base ∧ 𝑅 ∈ dom Base) → (Base‘𝑅) ∈ V)
14	13	funfni 5385	. . . . . . . 8 ⊢ ((Base Fn V ∧ 𝑅 ∈ V) → (Base‘𝑅) ∈ V)
15	11, 12, 14	sylancr 414	. . . . . . 7 ⊢ (𝑅 ∈ Rng → (Base‘𝑅) ∈ V)
16	15	pwexd 4233	. . . . . 6 ⊢ (𝑅 ∈ Rng → 𝒫 (Base‘𝑅) ∈ V)
17		rabexg 4195	. . . . . 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 5686	. . . 4 ⊢ (𝑅 ∈ Rng → (SubRng‘𝑅) = {𝑠 ∈ 𝒫 (Base‘𝑅) ∣ (𝑅 ↾s 𝑠) ∈ Rng})
20	19	eleq2d 2276	. . 3 ⊢ (𝑅 ∈ Rng → (𝐴 ∈ (SubRng‘𝑅) ↔ 𝐴 ∈ {𝑠 ∈ 𝒫 (Base‘𝑅) ∣ (𝑅 ↾s 𝑠) ∈ Rng}))
21		oveq2 5965	. . . . . 6 ⊢ (𝑠 = 𝐴 → (𝑅 ↾s 𝑠) = (𝑅 ↾s 𝐴))
22	21	eleq1d 2275	. . . . 5 ⊢ (𝑠 = 𝐴 → ((𝑅 ↾s 𝑠) ∈ Rng ↔ (𝑅 ↾s 𝐴) ∈ Rng))
23	22	elrab 2933	. . . 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 2210	. . . . . . . 8 ⊢ (Base‘𝑅) = 𝐵
26	25	sseq2i 3224	. . . . . . 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 4208	. . . . . . . 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 2177  {crab 2489  Vcvv 2773   ⊆ wss 3170  𝒫 cpw 3621   Fn wfn 5275  ‘cfv 5280  (class class class)co 5957  Basecbs 12907   ↾_s cress 12908  Rngcrng 13769  SubRngcsubrng 14034

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 2179  ax-14 2180  ax-ext 2188  ax-sep 4170  ax-pow 4226  ax-pr 4261  ax-un 4488  ax-cnex 8036  ax-resscn 8037  ax-1re 8039  ax-addrcl 8042

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 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-un 3174  df-in 3176  df-ss 3183  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3857  df-int 3892  df-br 4052  df-opab 4114  df-mpt 4115  df-id 4348  df-xp 4689  df-rel 4690  df-cnv 4691  df-co 4692  df-dm 4693  df-rn 4694  df-res 4695  df-ima 4696  df-iota 5241  df-fun 5282  df-fn 5283  df-fv 5288  df-ov 5960  df-inn 9057  df-ndx 12910  df-slot 12911  df-base 12913  df-subrng 14035

This theorem is referenced by:  subrngss  14037  subrngid  14038  subrngrng  14039  subrngrcl  14040  issubrng2  14047  subsubrng  14051  subrngpropd  14053  rng2idlsubrng  14354