Theorem subrngss 20451

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 20450	. 2 ⊢ (𝐴 ∈ (SubRng‘𝑅) ↔ (𝑅 ∈ Rng ∧ (𝑅 ↾s 𝐴) ∈ Rng ∧ 𝐴 ⊆ 𝐵))
3	2	simp3bi 1147	1 ⊢ (𝐴 ∈ (SubRng‘𝑅) → 𝐴 ⊆ 𝐵)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109   ⊆ wss 3905  ‘cfv 6486  (class class class)co 7353  Basecbs 17138   ↾_s cress 17159  Rngcrng 20055  SubRngcsubrng 20448

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 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374

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 3397  df-v 3440  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-iota 6442  df-fun 6488  df-fv 6494  df-ov 7356  df-subrng 20449

This theorem is referenced by:  subrngsubg  20455  subrngmre  20465  subsubrng  20466  rhmimasubrnglem  20468  rhmimasubrng  20469