Theorem subsubrng2 14187

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 14186	. . 3 ⊢ (𝐴 ∈ (SubRng‘𝑅) → (𝑎 ∈ (SubRng‘𝑆) ↔ (𝑎 ∈ (SubRng‘𝑅) ∧ 𝑎 ⊆ 𝐴)))
3		elin 3387	. . . 4 ⊢ (𝑎 ∈ ((SubRng‘𝑅) ∩ 𝒫 𝐴) ↔ (𝑎 ∈ (SubRng‘𝑅) ∧ 𝑎 ∈ 𝒫 𝐴))
4		velpw 3656	. . . . 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 2227	1 ⊢ (𝐴 ∈ (SubRng‘𝑅) → (SubRng‘𝑆) = ((SubRng‘𝑅) ∩ 𝒫 𝐴))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1395   ∈ wcel 2200   ∩ cin 3196   ⊆ wss 3197  𝒫 cpw 3649  ‘cfv 5318  (class class class)co 6007   ↾_s cress 13041  SubRngcsubrng 14169

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8098  ax-resscn 8099  ax-1re 8101  ax-addrcl 8104

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-fv 5326  df-ov 6010  df-oprab 6011  df-mpo 6012  df-inn 9119  df-2 9177  df-3 9178  df-ndx 13043  df-slot 13044  df-base 13046  df-sets 13047  df-iress 13048  df-plusg 13131  df-mulr 13132  df-subg 13715  df-abl 13832  df-rng 13904  df-subrng 14170

This theorem is referenced by: (None)