Theorem subrngid 20494

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 17183	. . 3 ⊢ (𝑅 ∈ Rng → (𝑅 ↾s 𝐵) = 𝑅)
4	3, 1	eqeltrd 2837	. 2 ⊢ (𝑅 ∈ Rng → (𝑅 ↾s 𝐵) ∈ Rng)
5		ssidd 3959	. 2 ⊢ (𝑅 ∈ Rng → 𝐵 ⊆ 𝐵)
6	2	issubrng 20492	. 2 ⊢ (𝐵 ∈ (SubRng‘𝑅) ↔ (𝑅 ∈ Rng ∧ (𝑅 ↾s 𝐵) ∈ Rng ∧ 𝐵 ⊆ 𝐵))
7	1, 4, 5, 6	syl3anbrc 1345	1 ⊢ (𝑅 ∈ Rng → 𝐵 ∈ (SubRng‘𝑅))

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114   ⊆ wss 3903  ‘cfv 6500  (class class class)co 7368  Basecbs 17148   ↾_s cress 17169  Rngcrng 20099  SubRngcsubrng 20490

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-sbc 3743  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fv 6508  df-ov 7371  df-oprab 7372  df-mpo 7373  df-ress 17170  df-subrng 20491

This theorem is referenced by:  subrngmre  20507