Theorem subrngid 20528

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 17212	. . 3 ⊢ (𝑅 ∈ Rng → (𝑅 ↾s 𝐵) = 𝑅)
4	3, 1	eqeltrd 2840	. 2 ⊢ (𝑅 ∈ Rng → (𝑅 ↾s 𝐵) ∈ Rng)
5		ssidd 3945	. 2 ⊢ (𝑅 ∈ Rng → 𝐵 ⊆ 𝐵)
6	2	issubrng 20526	. 2 ⊢ (𝐵 ∈ (SubRng‘𝑅) ↔ (𝑅 ∈ Rng ∧ (𝑅 ↾s 𝐵) ∈ Rng ∧ 𝐵 ⊆ 𝐵))
7	1, 4, 5, 6	syl3anbrc 1350	1 ⊢ (𝑅 ∈ Rng → 𝐵 ∈ (SubRng‘𝑅))

Syntax hints:   → wi 4   = wceq 1547   ∈ wcel 2119   ⊆ wss 3890  ‘cfv 6492  (class class class)co 7363  Basecbs 17177   ↾_s cress 17198  Rngcrng 20131  SubRngcsubrng 20524

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-sbc 3731  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-br 5080  df-opab 5142  df-mpt 5161  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6448  df-fun 6494  df-fv 6500  df-ov 7366  df-oprab 7367  df-mpo 7368  df-ress 17199  df-subrng 20525

This theorem is referenced by:  subrngmre  20541