Theorem subrngss 20629

Description: A subring is a subset. (Contributed by AV, 14-Feb-2025.)

Hypothesis

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

Assertion

Ref	Expression
subrngss	⊢ (𝐴 ∈ (SubRng‘𝑅) → 𝐴 ⊆ 𝐵)

Proof of Theorem subrngss

Step	Hyp	Ref	Expression
1		subrngss.1	. . 3 ⊢ 𝐵 = (Base‘𝑅)
2	1	issubrng 20628	. 2 ⊢ (𝐴 ∈ (SubRng‘𝑅) ↔ (𝑅 ∈ Rng ∧ (𝑅 ↾s 𝐴) ∈ Rng ∧ 𝐴 ⊆ 𝐵))
3	2	simp3bi 1163	1 ⊢ (𝐴 ∈ (SubRng‘𝑅) → 𝐴 ⊆ 𝐵)

Syntax hints:   → wi 4   = wceq 1567   ∈ wcel 2149   ⊆ wss 3913  ‘cfv 6533  (class class class)co 7408  Basecbs 17265   ↾_s cress 17286  Rngcrng 20226  SubRngcsubrng 20626

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5258  ax-nul 5268  ax-pow 5334  ax-pr 5402

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6489  df-fun 6535  df-fv 6541  df-ov 7411  df-subrng 20627

This theorem is referenced by:  subrngsubg  20633  subrngmre  20643  subsubrng  20644  rhmimasubrnglem  20646  rhmimasubrng  20647