Theorem subsubrng2 13559

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 13558	. . 3 ⊢ (𝐴 ∈ (SubRng‘𝑅) → (𝑎 ∈ (SubRng‘𝑆) ↔ (𝑎 ∈ (SubRng‘𝑅) ∧ 𝑎 ⊆ 𝐴)))
3		elin 3333	. . . 4 ⊢ (𝑎 ∈ ((SubRng‘𝑅) ∩ 𝒫 𝐴) ↔ (𝑎 ∈ (SubRng‘𝑅) ∧ 𝑎 ∈ 𝒫 𝐴))
4		velpw 3597	. . . . 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 2187	1 ⊢ (𝐴 ∈ (SubRng‘𝑅) → (SubRng‘𝑆) = ((SubRng‘𝑅) ∩ 𝒫 𝐴))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1364   ∈ wcel 2160   ∩ cin 3143   ⊆ wss 3144  𝒫 cpw 3590  ‘cfv 5235  (class class class)co 5895   ↾_s cress 12512  SubRngcsubrng 13541

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-cnex 7931  ax-resscn 7932  ax-1re 7934  ax-addrcl 7937

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-fv 5243  df-ov 5898  df-oprab 5899  df-mpo 5900  df-inn 8949  df-2 9007  df-3 9008  df-ndx 12514  df-slot 12515  df-base 12517  df-sets 12518  df-iress 12519  df-plusg 12599  df-mulr 12600  df-subg 13106  df-abl 13223  df-rng 13284  df-subrng 13542

This theorem is referenced by: (None)