Theorem subsubrng2 14021

Description: The set of subrings of a subring are the smaller subrings. (Contributed by AV, 15-Feb-2025.)

Hypothesis

Ref	Expression
subsubrng.s	⊢ 𝑆 = (𝑅 ↾s 𝐴)

Assertion

Ref	Expression
subsubrng2	⊢ (𝐴 ∈ (SubRng‘𝑅) → (SubRng‘𝑆) = ((SubRng‘𝑅) ∩ 𝒫 𝐴))

Proof of Theorem subsubrng2

Dummy variable 𝑎 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		subsubrng.s	. . . 4 ⊢ 𝑆 = (𝑅 ↾s 𝐴)
2	1	subsubrng 14020	. . 3 ⊢ (𝐴 ∈ (SubRng‘𝑅) → (𝑎 ∈ (SubRng‘𝑆) ↔ (𝑎 ∈ (SubRng‘𝑅) ∧ 𝑎 ⊆ 𝐴)))
3		elin 3357	. . . 4 ⊢ (𝑎 ∈ ((SubRng‘𝑅) ∩ 𝒫 𝐴) ↔ (𝑎 ∈ (SubRng‘𝑅) ∧ 𝑎 ∈ 𝒫 𝐴))
4		velpw 3624	. . . . 5 ⊢ (𝑎 ∈ 𝒫 𝐴 ↔ 𝑎 ⊆ 𝐴)
5	4	anbi2i 457	. . . 4 ⊢ ((𝑎 ∈ (SubRng‘𝑅) ∧ 𝑎 ∈ 𝒫 𝐴) ↔ (𝑎 ∈ (SubRng‘𝑅) ∧ 𝑎 ⊆ 𝐴))
6	3, 5	bitr2i 185	. . 3 ⊢ ((𝑎 ∈ (SubRng‘𝑅) ∧ 𝑎 ⊆ 𝐴) ↔ 𝑎 ∈ ((SubRng‘𝑅) ∩ 𝒫 𝐴))
7	2, 6	bitrdi 196	. 2 ⊢ (𝐴 ∈ (SubRng‘𝑅) → (𝑎 ∈ (SubRng‘𝑆) ↔ 𝑎 ∈ ((SubRng‘𝑅) ∩ 𝒫 𝐴)))
8	7	eqrdv 2204	1 ⊢ (𝐴 ∈ (SubRng‘𝑅) → (SubRng‘𝑆) = ((SubRng‘𝑅) ∩ 𝒫 𝐴))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1373   ∈ wcel 2177   ∩ cin 3166   ⊆ wss 3167  𝒫 cpw 3617  ‘cfv 5276  (class class class)co 5951   ↾_s cress 12877  SubRngcsubrng 14003

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-sep 4166  ax-pow 4222  ax-pr 4257  ax-un 4484  ax-setind 4589  ax-cnex 8023  ax-resscn 8024  ax-1re 8026  ax-addrcl 8029

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-ral 2490  df-rex 2491  df-rab 2494  df-v 2775  df-sbc 3000  df-csb 3095  df-dif 3169  df-un 3171  df-in 3173  df-ss 3180  df-nul 3462  df-pw 3619  df-sn 3640  df-pr 3641  df-op 3643  df-uni 3853  df-int 3888  df-br 4048  df-opab 4110  df-mpt 4111  df-id 4344  df-xp 4685  df-rel 4686  df-cnv 4687  df-co 4688  df-dm 4689  df-rn 4690  df-res 4691  df-ima 4692  df-iota 5237  df-fun 5278  df-fn 5279  df-fv 5284  df-ov 5954  df-oprab 5955  df-mpo 5956  df-inn 9044  df-2 9102  df-3 9103  df-ndx 12879  df-slot 12880  df-base 12882  df-sets 12883  df-iress 12884  df-plusg 12966  df-mulr 12967  df-subg 13550  df-abl 13667  df-rng 13739  df-subrng 14004

This theorem is referenced by: (None)