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Theorem issubrng 20547
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 ∧ (𝑅s 𝐴) ∈ Rng ∧ 𝐴𝐵))

Proof of Theorem issubrng
Dummy variables 𝑤 𝑠 𝑟 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-subrng 20546 . . 3 SubRng = (𝑤 ∈ Rng ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤s 𝑠) ∈ Rng})
21mptrcl 7025 . 2 (𝐴 ∈ (SubRng‘𝑅) → 𝑅 ∈ Rng)
3 simp1 1137 . 2 ((𝑅 ∈ Rng ∧ (𝑅s 𝐴) ∈ Rng ∧ 𝐴𝐵) → 𝑅 ∈ Rng)
4 fveq2 6906 . . . . . . 7 (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
54pweqd 4617 . . . . . 6 (𝑟 = 𝑅 → 𝒫 (Base‘𝑟) = 𝒫 (Base‘𝑅))
6 oveq1 7438 . . . . . . 7 (𝑟 = 𝑅 → (𝑟s 𝑠) = (𝑅s 𝑠))
76eleq1d 2826 . . . . . 6 (𝑟 = 𝑅 → ((𝑟s 𝑠) ∈ Rng ↔ (𝑅s 𝑠) ∈ Rng))
85, 7rabeqbidv 3455 . . . . 5 (𝑟 = 𝑅 → {𝑠 ∈ 𝒫 (Base‘𝑟) ∣ (𝑟s 𝑠) ∈ Rng} = {𝑠 ∈ 𝒫 (Base‘𝑅) ∣ (𝑅s 𝑠) ∈ Rng})
9 df-subrng 20546 . . . . 5 SubRng = (𝑟 ∈ Rng ↦ {𝑠 ∈ 𝒫 (Base‘𝑟) ∣ (𝑟s 𝑠) ∈ Rng})
10 fvex 6919 . . . . . . 7 (Base‘𝑅) ∈ V
1110pwex 5380 . . . . . 6 𝒫 (Base‘𝑅) ∈ V
1211rabex 5339 . . . . 5 {𝑠 ∈ 𝒫 (Base‘𝑅) ∣ (𝑅s 𝑠) ∈ Rng} ∈ V
138, 9, 12fvmpt 7016 . . . 4 (𝑅 ∈ Rng → (SubRng‘𝑅) = {𝑠 ∈ 𝒫 (Base‘𝑅) ∣ (𝑅s 𝑠) ∈ Rng})
1413eleq2d 2827 . . 3 (𝑅 ∈ Rng → (𝐴 ∈ (SubRng‘𝑅) ↔ 𝐴 ∈ {𝑠 ∈ 𝒫 (Base‘𝑅) ∣ (𝑅s 𝑠) ∈ Rng}))
15 oveq2 7439 . . . . . 6 (𝑠 = 𝐴 → (𝑅s 𝑠) = (𝑅s 𝐴))
1615eleq1d 2826 . . . . 5 (𝑠 = 𝐴 → ((𝑅s 𝑠) ∈ Rng ↔ (𝑅s 𝐴) ∈ Rng))
1716elrab 3692 . . . 4 (𝐴 ∈ {𝑠 ∈ 𝒫 (Base‘𝑅) ∣ (𝑅s 𝑠) ∈ Rng} ↔ (𝐴 ∈ 𝒫 (Base‘𝑅) ∧ (𝑅s 𝐴) ∈ Rng))
18 issubrng.b . . . . . . . . 9 𝐵 = (Base‘𝑅)
1918eqcomi 2746 . . . . . . . 8 (Base‘𝑅) = 𝐵
2019sseq2i 4013 . . . . . . 7 (𝐴 ⊆ (Base‘𝑅) ↔ 𝐴𝐵)
2120anbi2i 623 . . . . . 6 (((𝑅s 𝐴) ∈ Rng ∧ 𝐴 ⊆ (Base‘𝑅)) ↔ ((𝑅s 𝐴) ∈ Rng ∧ 𝐴𝐵))
22 ibar 528 . . . . . 6 (𝑅 ∈ Rng → (((𝑅s 𝐴) ∈ Rng ∧ 𝐴𝐵) ↔ (𝑅 ∈ Rng ∧ ((𝑅s 𝐴) ∈ Rng ∧ 𝐴𝐵))))
2321, 22bitrid 283 . . . . 5 (𝑅 ∈ Rng → (((𝑅s 𝐴) ∈ Rng ∧ 𝐴 ⊆ (Base‘𝑅)) ↔ (𝑅 ∈ Rng ∧ ((𝑅s 𝐴) ∈ Rng ∧ 𝐴𝐵))))
2410elpw2 5334 . . . . . 6 (𝐴 ∈ 𝒫 (Base‘𝑅) ↔ 𝐴 ⊆ (Base‘𝑅))
2524anbi2ci 625 . . . . 5 ((𝐴 ∈ 𝒫 (Base‘𝑅) ∧ (𝑅s 𝐴) ∈ Rng) ↔ ((𝑅s 𝐴) ∈ Rng ∧ 𝐴 ⊆ (Base‘𝑅)))
26 3anass 1095 . . . . 5 ((𝑅 ∈ Rng ∧ (𝑅s 𝐴) ∈ Rng ∧ 𝐴𝐵) ↔ (𝑅 ∈ Rng ∧ ((𝑅s 𝐴) ∈ Rng ∧ 𝐴𝐵)))
2723, 25, 263bitr4g 314 . . . 4 (𝑅 ∈ Rng → ((𝐴 ∈ 𝒫 (Base‘𝑅) ∧ (𝑅s 𝐴) ∈ Rng) ↔ (𝑅 ∈ Rng ∧ (𝑅s 𝐴) ∈ Rng ∧ 𝐴𝐵)))
2817, 27bitrid 283 . . 3 (𝑅 ∈ Rng → (𝐴 ∈ {𝑠 ∈ 𝒫 (Base‘𝑅) ∣ (𝑅s 𝑠) ∈ Rng} ↔ (𝑅 ∈ Rng ∧ (𝑅s 𝐴) ∈ Rng ∧ 𝐴𝐵)))
2914, 28bitrd 279 . 2 (𝑅 ∈ Rng → (𝐴 ∈ (SubRng‘𝑅) ↔ (𝑅 ∈ Rng ∧ (𝑅s 𝐴) ∈ Rng ∧ 𝐴𝐵)))
302, 3, 29pm5.21nii 378 1 (𝐴 ∈ (SubRng‘𝑅) ↔ (𝑅 ∈ Rng ∧ (𝑅s 𝐴) ∈ Rng ∧ 𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  {crab 3436  wss 3951  𝒫 cpw 4600  cfv 6561  (class class class)co 7431  Basecbs 17247  s cress 17274  Rngcrng 20149  SubRngcsubrng 20545
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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fv 6569  df-ov 7434  df-subrng 20546
This theorem is referenced by:  subrngss  20548  subrngid  20549  subrngrng  20550  subrngrcl  20551  issubrng2  20558  subsubrng  20563  subrngpropd  20568  subrgsubrng  20578  rng2idlsubrng  21275
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