Theorem subrngid 20458

Description: Every non-unital ring is a subring of itself. (Contributed by AV, 14-Feb-2025.)

Hypothesis

Ref	Expression
subrngss.1	⊢ 𝐵 = (Base‘𝑅)

Assertion

Ref	Expression
subrngid	⊢ (𝑅 ∈ Rng → 𝐵 ∈ (SubRng‘𝑅))

Proof of Theorem subrngid

Step	Hyp	Ref	Expression
1		id 22	. 2 ⊢ (𝑅 ∈ Rng → 𝑅 ∈ Rng)
2		subrngss.1	. . . 4 ⊢ 𝐵 = (Base‘𝑅)
3	2	ressid 17214	. . 3 ⊢ (𝑅 ∈ Rng → (𝑅 ↾s 𝐵) = 𝑅)
4	3, 1	eqeltrd 2828	. 2 ⊢ (𝑅 ∈ Rng → (𝑅 ↾s 𝐵) ∈ Rng)
5		ssidd 3970	. 2 ⊢ (𝑅 ∈ Rng → 𝐵 ⊆ 𝐵)
6	2	issubrng 20456	. 2 ⊢ (𝐵 ∈ (SubRng‘𝑅) ↔ (𝑅 ∈ Rng ∧ (𝑅 ↾s 𝐵) ∈ Rng ∧ 𝐵 ⊆ 𝐵))
7	1, 4, 5, 6	syl3anbrc 1344	1 ⊢ (𝑅 ∈ Rng → 𝐵 ∈ (SubRng‘𝑅))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109   ⊆ wss 3914  ‘cfv 6511  (class class class)co 7387  Basecbs 17179   ↾_s cress 17200  Rngcrng 20061  SubRngcsubrng 20454

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 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387

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 3406  df-v 3449  df-sbc 3754  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fv 6519  df-ov 7390  df-oprab 7391  df-mpo 7392  df-ress 17201  df-subrng 20455

This theorem is referenced by:  subrngmre  20471