Theorem subrngid 20498

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 17228	. . 3 ⊢ (𝑅 ∈ Rng → (𝑅 ↾s 𝐵) = 𝑅)
4	3, 1	eqeltrd 2825	. 2 ⊢ (𝑅 ∈ Rng → (𝑅 ↾s 𝐵) ∈ Rng)
5		ssidd 4000	. 2 ⊢ (𝑅 ∈ Rng → 𝐵 ⊆ 𝐵)
6	2	issubrng 20496	. 2 ⊢ (𝐵 ∈ (SubRng‘𝑅) ↔ (𝑅 ∈ Rng ∧ (𝑅 ↾s 𝐵) ∈ Rng ∧ 𝐵 ⊆ 𝐵))
7	1, 4, 5, 6	syl3anbrc 1340	1 ⊢ (𝑅 ∈ Rng → 𝐵 ∈ (SubRng‘𝑅))

Syntax hints:   → wi 4   = wceq 1533   ∈ wcel 2098   ⊆ wss 3944  ‘cfv 6549  (class class class)co 7419  Basecbs 17183   ↾_s cress 17212  Rngcrng 20104  SubRngcsubrng 20494

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-sbc 3774  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6501  df-fun 6551  df-fv 6557  df-ov 7422  df-oprab 7423  df-mpo 7424  df-ress 17213  df-subrng 20495

This theorem is referenced by:  subrngmre  20511