Theorem ssufl 23069

Description: If 𝑌 is a subset of 𝑋 and filters extend to ultrafilters in 𝑋, then they still do in 𝑌. (Contributed by Mario Carneiro, 26-Aug-2015.)

Assertion

Ref	Expression
ssufl	⊢ ((𝑋 ∈ UFL ∧ 𝑌 ⊆ 𝑋) → 𝑌 ∈ UFL)

Proof of Theorem ssufl

Dummy variables 𝑓 𝑔 𝑢 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpll 764	. . . . 5 ⊢ (((𝑋 ∈ UFL ∧ 𝑌 ⊆ 𝑋) ∧ 𝑓 ∈ (Fil‘𝑌)) → 𝑋 ∈ UFL)
2		filfbas 22999	. . . . . . . 8 ⊢ (𝑓 ∈ (Fil‘𝑌) → 𝑓 ∈ (fBas‘𝑌))
3	2	adantl 482	. . . . . . 7 ⊢ (((𝑋 ∈ UFL ∧ 𝑌 ⊆ 𝑋) ∧ 𝑓 ∈ (Fil‘𝑌)) → 𝑓 ∈ (fBas‘𝑌))
4		filsspw 23002	. . . . . . . . 9 ⊢ (𝑓 ∈ (Fil‘𝑌) → 𝑓 ⊆ 𝒫 𝑌)
5	4	adantl 482	. . . . . . . 8 ⊢ (((𝑋 ∈ UFL ∧ 𝑌 ⊆ 𝑋) ∧ 𝑓 ∈ (Fil‘𝑌)) → 𝑓 ⊆ 𝒫 𝑌)
6		simplr 766	. . . . . . . . 9 ⊢ (((𝑋 ∈ UFL ∧ 𝑌 ⊆ 𝑋) ∧ 𝑓 ∈ (Fil‘𝑌)) → 𝑌 ⊆ 𝑋)
7	6	sspwd 4548	. . . . . . . 8 ⊢ (((𝑋 ∈ UFL ∧ 𝑌 ⊆ 𝑋) ∧ 𝑓 ∈ (Fil‘𝑌)) → 𝒫 𝑌 ⊆ 𝒫 𝑋)
8	5, 7	sstrd 3931	. . . . . . 7 ⊢ (((𝑋 ∈ UFL ∧ 𝑌 ⊆ 𝑋) ∧ 𝑓 ∈ (Fil‘𝑌)) → 𝑓 ⊆ 𝒫 𝑋)
9		fbasweak 23016	. . . . . . 7 ⊢ ((𝑓 ∈ (fBas‘𝑌) ∧ 𝑓 ⊆ 𝒫 𝑋 ∧ 𝑋 ∈ UFL) → 𝑓 ∈ (fBas‘𝑋))
10	3, 8, 1, 9	syl3anc 1370	. . . . . 6 ⊢ (((𝑋 ∈ UFL ∧ 𝑌 ⊆ 𝑋) ∧ 𝑓 ∈ (Fil‘𝑌)) → 𝑓 ∈ (fBas‘𝑋))
11		fgcl 23029	. . . . . 6 ⊢ (𝑓 ∈ (fBas‘𝑋) → (𝑋filGen𝑓) ∈ (Fil‘𝑋))
12	10, 11	syl 17	. . . . 5 ⊢ (((𝑋 ∈ UFL ∧ 𝑌 ⊆ 𝑋) ∧ 𝑓 ∈ (Fil‘𝑌)) → (𝑋filGen𝑓) ∈ (Fil‘𝑋))
13		ufli 23065	. . . . 5 ⊢ ((𝑋 ∈ UFL ∧ (𝑋filGen𝑓) ∈ (Fil‘𝑋)) → ∃𝑢 ∈ (UFil‘𝑋)(𝑋filGen𝑓) ⊆ 𝑢)
14	1, 12, 13	syl2anc 584	. . . 4 ⊢ (((𝑋 ∈ UFL ∧ 𝑌 ⊆ 𝑋) ∧ 𝑓 ∈ (Fil‘𝑌)) → ∃𝑢 ∈ (UFil‘𝑋)(𝑋filGen𝑓) ⊆ 𝑢)
15		ssfg 23023	. . . . . . . . . 10 ⊢ (𝑓 ∈ (fBas‘𝑋) → 𝑓 ⊆ (𝑋filGen𝑓))
16	10, 15	syl 17	. . . . . . . . 9 ⊢ (((𝑋 ∈ UFL ∧ 𝑌 ⊆ 𝑋) ∧ 𝑓 ∈ (Fil‘𝑌)) → 𝑓 ⊆ (𝑋filGen𝑓))
17	16	adantr 481	. . . . . . . 8 ⊢ ((((𝑋 ∈ UFL ∧ 𝑌 ⊆ 𝑋) ∧ 𝑓 ∈ (Fil‘𝑌)) ∧ (𝑢 ∈ (UFil‘𝑋) ∧ (𝑋filGen𝑓) ⊆ 𝑢)) → 𝑓 ⊆ (𝑋filGen𝑓))
18		simprr 770	. . . . . . . 8 ⊢ ((((𝑋 ∈ UFL ∧ 𝑌 ⊆ 𝑋) ∧ 𝑓 ∈ (Fil‘𝑌)) ∧ (𝑢 ∈ (UFil‘𝑋) ∧ (𝑋filGen𝑓) ⊆ 𝑢)) → (𝑋filGen𝑓) ⊆ 𝑢)
19	17, 18	sstrd 3931	. . . . . . 7 ⊢ ((((𝑋 ∈ UFL ∧ 𝑌 ⊆ 𝑋) ∧ 𝑓 ∈ (Fil‘𝑌)) ∧ (𝑢 ∈ (UFil‘𝑋) ∧ (𝑋filGen𝑓) ⊆ 𝑢)) → 𝑓 ⊆ 𝑢)
20		filtop 23006	. . . . . . . 8 ⊢ (𝑓 ∈ (Fil‘𝑌) → 𝑌 ∈ 𝑓)
21	20	ad2antlr 724	. . . . . . 7 ⊢ ((((𝑋 ∈ UFL ∧ 𝑌 ⊆ 𝑋) ∧ 𝑓 ∈ (Fil‘𝑌)) ∧ (𝑢 ∈ (UFil‘𝑋) ∧ (𝑋filGen𝑓) ⊆ 𝑢)) → 𝑌 ∈ 𝑓)
22	19, 21	sseldd 3922	. . . . . 6 ⊢ ((((𝑋 ∈ UFL ∧ 𝑌 ⊆ 𝑋) ∧ 𝑓 ∈ (Fil‘𝑌)) ∧ (𝑢 ∈ (UFil‘𝑋) ∧ (𝑋filGen𝑓) ⊆ 𝑢)) → 𝑌 ∈ 𝑢)
23		simprl 768	. . . . . . 7 ⊢ ((((𝑋 ∈ UFL ∧ 𝑌 ⊆ 𝑋) ∧ 𝑓 ∈ (Fil‘𝑌)) ∧ (𝑢 ∈ (UFil‘𝑋) ∧ (𝑋filGen𝑓) ⊆ 𝑢)) → 𝑢 ∈ (UFil‘𝑋))
24	6	adantr 481	. . . . . . 7 ⊢ ((((𝑋 ∈ UFL ∧ 𝑌 ⊆ 𝑋) ∧ 𝑓 ∈ (Fil‘𝑌)) ∧ (𝑢 ∈ (UFil‘𝑋) ∧ (𝑋filGen𝑓) ⊆ 𝑢)) → 𝑌 ⊆ 𝑋)
25		trufil 23061	. . . . . . 7 ⊢ ((𝑢 ∈ (UFil‘𝑋) ∧ 𝑌 ⊆ 𝑋) → ((𝑢 ↾t 𝑌) ∈ (UFil‘𝑌) ↔ 𝑌 ∈ 𝑢))
26	23, 24, 25	syl2anc 584	. . . . . 6 ⊢ ((((𝑋 ∈ UFL ∧ 𝑌 ⊆ 𝑋) ∧ 𝑓 ∈ (Fil‘𝑌)) ∧ (𝑢 ∈ (UFil‘𝑋) ∧ (𝑋filGen𝑓) ⊆ 𝑢)) → ((𝑢 ↾t 𝑌) ∈ (UFil‘𝑌) ↔ 𝑌 ∈ 𝑢))
27	22, 26	mpbird 256	. . . . 5 ⊢ ((((𝑋 ∈ UFL ∧ 𝑌 ⊆ 𝑋) ∧ 𝑓 ∈ (Fil‘𝑌)) ∧ (𝑢 ∈ (UFil‘𝑋) ∧ (𝑋filGen𝑓) ⊆ 𝑢)) → (𝑢 ↾t 𝑌) ∈ (UFil‘𝑌))
28	5	adantr 481	. . . . . . 7 ⊢ ((((𝑋 ∈ UFL ∧ 𝑌 ⊆ 𝑋) ∧ 𝑓 ∈ (Fil‘𝑌)) ∧ (𝑢 ∈ (UFil‘𝑋) ∧ (𝑋filGen𝑓) ⊆ 𝑢)) → 𝑓 ⊆ 𝒫 𝑌)
29		restid2 17141	. . . . . . 7 ⊢ ((𝑌 ∈ 𝑓 ∧ 𝑓 ⊆ 𝒫 𝑌) → (𝑓 ↾t 𝑌) = 𝑓)
30	21, 28, 29	syl2anc 584	. . . . . 6 ⊢ ((((𝑋 ∈ UFL ∧ 𝑌 ⊆ 𝑋) ∧ 𝑓 ∈ (Fil‘𝑌)) ∧ (𝑢 ∈ (UFil‘𝑋) ∧ (𝑋filGen𝑓) ⊆ 𝑢)) → (𝑓 ↾t 𝑌) = 𝑓)
31		ssrest 22327	. . . . . . 7 ⊢ ((𝑢 ∈ (UFil‘𝑋) ∧ 𝑓 ⊆ 𝑢) → (𝑓 ↾t 𝑌) ⊆ (𝑢 ↾t 𝑌))
32	23, 19, 31	syl2anc 584	. . . . . 6 ⊢ ((((𝑋 ∈ UFL ∧ 𝑌 ⊆ 𝑋) ∧ 𝑓 ∈ (Fil‘𝑌)) ∧ (𝑢 ∈ (UFil‘𝑋) ∧ (𝑋filGen𝑓) ⊆ 𝑢)) → (𝑓 ↾t 𝑌) ⊆ (𝑢 ↾t 𝑌))
33	30, 32	eqsstrrd 3960	. . . . 5 ⊢ ((((𝑋 ∈ UFL ∧ 𝑌 ⊆ 𝑋) ∧ 𝑓 ∈ (Fil‘𝑌)) ∧ (𝑢 ∈ (UFil‘𝑋) ∧ (𝑋filGen𝑓) ⊆ 𝑢)) → 𝑓 ⊆ (𝑢 ↾t 𝑌))
34		sseq2 3947	. . . . . 6 ⊢ (𝑔 = (𝑢 ↾t 𝑌) → (𝑓 ⊆ 𝑔 ↔ 𝑓 ⊆ (𝑢 ↾t 𝑌)))
35	34	rspcev 3561	. . . . 5 ⊢ (((𝑢 ↾t 𝑌) ∈ (UFil‘𝑌) ∧ 𝑓 ⊆ (𝑢 ↾t 𝑌)) → ∃𝑔 ∈ (UFil‘𝑌)𝑓 ⊆ 𝑔)
36	27, 33, 35	syl2anc 584	. . . 4 ⊢ ((((𝑋 ∈ UFL ∧ 𝑌 ⊆ 𝑋) ∧ 𝑓 ∈ (Fil‘𝑌)) ∧ (𝑢 ∈ (UFil‘𝑋) ∧ (𝑋filGen𝑓) ⊆ 𝑢)) → ∃𝑔 ∈ (UFil‘𝑌)𝑓 ⊆ 𝑔)
37	14, 36	rexlimddv 3220	. . 3 ⊢ (((𝑋 ∈ UFL ∧ 𝑌 ⊆ 𝑋) ∧ 𝑓 ∈ (Fil‘𝑌)) → ∃𝑔 ∈ (UFil‘𝑌)𝑓 ⊆ 𝑔)
38	37	ralrimiva 3103	. 2 ⊢ ((𝑋 ∈ UFL ∧ 𝑌 ⊆ 𝑋) → ∀𝑓 ∈ (Fil‘𝑌)∃𝑔 ∈ (UFil‘𝑌)𝑓 ⊆ 𝑔)
39		ssexg 5247	. . . 4 ⊢ ((𝑌 ⊆ 𝑋 ∧ 𝑋 ∈ UFL) → 𝑌 ∈ V)
40	39	ancoms 459	. . 3 ⊢ ((𝑋 ∈ UFL ∧ 𝑌 ⊆ 𝑋) → 𝑌 ∈ V)
41		isufl 23064	. . 3 ⊢ (𝑌 ∈ V → (𝑌 ∈ UFL ↔ ∀𝑓 ∈ (Fil‘𝑌)∃𝑔 ∈ (UFil‘𝑌)𝑓 ⊆ 𝑔))
42	40, 41	syl 17	. 2 ⊢ ((𝑋 ∈ UFL ∧ 𝑌 ⊆ 𝑋) → (𝑌 ∈ UFL ↔ ∀𝑓 ∈ (Fil‘𝑌)∃𝑔 ∈ (UFil‘𝑌)𝑓 ⊆ 𝑔))
43	38, 42	mpbird 256	1 ⊢ ((𝑋 ∈ UFL ∧ 𝑌 ⊆ 𝑋) → 𝑌 ∈ UFL)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106  ∀wral 3064  ∃wrex 3065  Vcvv 3432   ⊆ wss 3887  𝒫 cpw 4533  ‘cfv 6433  (class class class)co 7275   ↾_t crest 17131  fBascfbas 20585  filGencfg 20586  Filcfil 22996  UFilcufil 23050  UFLcufl 23051

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-1st 7831  df-2nd 7832  df-rest 17133  df-fbas 20594  df-fg 20595  df-fil 22997  df-ufil 23052  df-ufl 23053

This theorem is referenced by:  ufldom  23113