Theorem ssufl 22977

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 763	. . . . 5 ⊢ (((𝑋 ∈ UFL ∧ 𝑌 ⊆ 𝑋) ∧ 𝑓 ∈ (Fil‘𝑌)) → 𝑋 ∈ UFL)
2		filfbas 22907	. . . . . . . 8 ⊢ (𝑓 ∈ (Fil‘𝑌) → 𝑓 ∈ (fBas‘𝑌))
3	2	adantl 481	. . . . . . 7 ⊢ (((𝑋 ∈ UFL ∧ 𝑌 ⊆ 𝑋) ∧ 𝑓 ∈ (Fil‘𝑌)) → 𝑓 ∈ (fBas‘𝑌))
4		filsspw 22910	. . . . . . . . 9 ⊢ (𝑓 ∈ (Fil‘𝑌) → 𝑓 ⊆ 𝒫 𝑌)
5	4	adantl 481	. . . . . . . 8 ⊢ (((𝑋 ∈ UFL ∧ 𝑌 ⊆ 𝑋) ∧ 𝑓 ∈ (Fil‘𝑌)) → 𝑓 ⊆ 𝒫 𝑌)
6		simplr 765	. . . . . . . . 9 ⊢ (((𝑋 ∈ UFL ∧ 𝑌 ⊆ 𝑋) ∧ 𝑓 ∈ (Fil‘𝑌)) → 𝑌 ⊆ 𝑋)
7	6	sspwd 4545	. . . . . . . 8 ⊢ (((𝑋 ∈ UFL ∧ 𝑌 ⊆ 𝑋) ∧ 𝑓 ∈ (Fil‘𝑌)) → 𝒫 𝑌 ⊆ 𝒫 𝑋)
8	5, 7	sstrd 3927	. . . . . . 7 ⊢ (((𝑋 ∈ UFL ∧ 𝑌 ⊆ 𝑋) ∧ 𝑓 ∈ (Fil‘𝑌)) → 𝑓 ⊆ 𝒫 𝑋)
9		fbasweak 22924	. . . . . . 7 ⊢ ((𝑓 ∈ (fBas‘𝑌) ∧ 𝑓 ⊆ 𝒫 𝑋 ∧ 𝑋 ∈ UFL) → 𝑓 ∈ (fBas‘𝑋))
10	3, 8, 1, 9	syl3anc 1369	. . . . . 6 ⊢ (((𝑋 ∈ UFL ∧ 𝑌 ⊆ 𝑋) ∧ 𝑓 ∈ (Fil‘𝑌)) → 𝑓 ∈ (fBas‘𝑋))
11		fgcl 22937	. . . . . 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 22973	. . . . 5 ⊢ ((𝑋 ∈ UFL ∧ (𝑋filGen𝑓) ∈ (Fil‘𝑋)) → ∃𝑢 ∈ (UFil‘𝑋)(𝑋filGen𝑓) ⊆ 𝑢)
14	1, 12, 13	syl2anc 583	. . . 4 ⊢ (((𝑋 ∈ UFL ∧ 𝑌 ⊆ 𝑋) ∧ 𝑓 ∈ (Fil‘𝑌)) → ∃𝑢 ∈ (UFil‘𝑋)(𝑋filGen𝑓) ⊆ 𝑢)
15		ssfg 22931	. . . . . . . . . 10 ⊢ (𝑓 ∈ (fBas‘𝑋) → 𝑓 ⊆ (𝑋filGen𝑓))
16	10, 15	syl 17	. . . . . . . . 9 ⊢ (((𝑋 ∈ UFL ∧ 𝑌 ⊆ 𝑋) ∧ 𝑓 ∈ (Fil‘𝑌)) → 𝑓 ⊆ (𝑋filGen𝑓))
17	16	adantr 480	. . . . . . . 8 ⊢ ((((𝑋 ∈ UFL ∧ 𝑌 ⊆ 𝑋) ∧ 𝑓 ∈ (Fil‘𝑌)) ∧ (𝑢 ∈ (UFil‘𝑋) ∧ (𝑋filGen𝑓) ⊆ 𝑢)) → 𝑓 ⊆ (𝑋filGen𝑓))
18		simprr 769	. . . . . . . 8 ⊢ ((((𝑋 ∈ UFL ∧ 𝑌 ⊆ 𝑋) ∧ 𝑓 ∈ (Fil‘𝑌)) ∧ (𝑢 ∈ (UFil‘𝑋) ∧ (𝑋filGen𝑓) ⊆ 𝑢)) → (𝑋filGen𝑓) ⊆ 𝑢)
19	17, 18	sstrd 3927	. . . . . . 7 ⊢ ((((𝑋 ∈ UFL ∧ 𝑌 ⊆ 𝑋) ∧ 𝑓 ∈ (Fil‘𝑌)) ∧ (𝑢 ∈ (UFil‘𝑋) ∧ (𝑋filGen𝑓) ⊆ 𝑢)) → 𝑓 ⊆ 𝑢)
20		filtop 22914	. . . . . . . 8 ⊢ (𝑓 ∈ (Fil‘𝑌) → 𝑌 ∈ 𝑓)
21	20	ad2antlr 723	. . . . . . 7 ⊢ ((((𝑋 ∈ UFL ∧ 𝑌 ⊆ 𝑋) ∧ 𝑓 ∈ (Fil‘𝑌)) ∧ (𝑢 ∈ (UFil‘𝑋) ∧ (𝑋filGen𝑓) ⊆ 𝑢)) → 𝑌 ∈ 𝑓)
22	19, 21	sseldd 3918	. . . . . 6 ⊢ ((((𝑋 ∈ UFL ∧ 𝑌 ⊆ 𝑋) ∧ 𝑓 ∈ (Fil‘𝑌)) ∧ (𝑢 ∈ (UFil‘𝑋) ∧ (𝑋filGen𝑓) ⊆ 𝑢)) → 𝑌 ∈ 𝑢)
23		simprl 767	. . . . . . 7 ⊢ ((((𝑋 ∈ UFL ∧ 𝑌 ⊆ 𝑋) ∧ 𝑓 ∈ (Fil‘𝑌)) ∧ (𝑢 ∈ (UFil‘𝑋) ∧ (𝑋filGen𝑓) ⊆ 𝑢)) → 𝑢 ∈ (UFil‘𝑋))
24	6	adantr 480	. . . . . . 7 ⊢ ((((𝑋 ∈ UFL ∧ 𝑌 ⊆ 𝑋) ∧ 𝑓 ∈ (Fil‘𝑌)) ∧ (𝑢 ∈ (UFil‘𝑋) ∧ (𝑋filGen𝑓) ⊆ 𝑢)) → 𝑌 ⊆ 𝑋)
25		trufil 22969	. . . . . . 7 ⊢ ((𝑢 ∈ (UFil‘𝑋) ∧ 𝑌 ⊆ 𝑋) → ((𝑢 ↾t 𝑌) ∈ (UFil‘𝑌) ↔ 𝑌 ∈ 𝑢))
26	23, 24, 25	syl2anc 583	. . . . . 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 480	. . . . . . 7 ⊢ ((((𝑋 ∈ UFL ∧ 𝑌 ⊆ 𝑋) ∧ 𝑓 ∈ (Fil‘𝑌)) ∧ (𝑢 ∈ (UFil‘𝑋) ∧ (𝑋filGen𝑓) ⊆ 𝑢)) → 𝑓 ⊆ 𝒫 𝑌)
29		restid2 17058	. . . . . . 7 ⊢ ((𝑌 ∈ 𝑓 ∧ 𝑓 ⊆ 𝒫 𝑌) → (𝑓 ↾t 𝑌) = 𝑓)
30	21, 28, 29	syl2anc 583	. . . . . 6 ⊢ ((((𝑋 ∈ UFL ∧ 𝑌 ⊆ 𝑋) ∧ 𝑓 ∈ (Fil‘𝑌)) ∧ (𝑢 ∈ (UFil‘𝑋) ∧ (𝑋filGen𝑓) ⊆ 𝑢)) → (𝑓 ↾t 𝑌) = 𝑓)
31		ssrest 22235	. . . . . . 7 ⊢ ((𝑢 ∈ (UFil‘𝑋) ∧ 𝑓 ⊆ 𝑢) → (𝑓 ↾t 𝑌) ⊆ (𝑢 ↾t 𝑌))
32	23, 19, 31	syl2anc 583	. . . . . 6 ⊢ ((((𝑋 ∈ UFL ∧ 𝑌 ⊆ 𝑋) ∧ 𝑓 ∈ (Fil‘𝑌)) ∧ (𝑢 ∈ (UFil‘𝑋) ∧ (𝑋filGen𝑓) ⊆ 𝑢)) → (𝑓 ↾t 𝑌) ⊆ (𝑢 ↾t 𝑌))
33	30, 32	eqsstrrd 3956	. . . . 5 ⊢ ((((𝑋 ∈ UFL ∧ 𝑌 ⊆ 𝑋) ∧ 𝑓 ∈ (Fil‘𝑌)) ∧ (𝑢 ∈ (UFil‘𝑋) ∧ (𝑋filGen𝑓) ⊆ 𝑢)) → 𝑓 ⊆ (𝑢 ↾t 𝑌))
34		sseq2 3943	. . . . . 6 ⊢ (𝑔 = (𝑢 ↾t 𝑌) → (𝑓 ⊆ 𝑔 ↔ 𝑓 ⊆ (𝑢 ↾t 𝑌)))
35	34	rspcev 3552	. . . . 5 ⊢ (((𝑢 ↾t 𝑌) ∈ (UFil‘𝑌) ∧ 𝑓 ⊆ (𝑢 ↾t 𝑌)) → ∃𝑔 ∈ (UFil‘𝑌)𝑓 ⊆ 𝑔)
36	27, 33, 35	syl2anc 583	. . . 4 ⊢ ((((𝑋 ∈ UFL ∧ 𝑌 ⊆ 𝑋) ∧ 𝑓 ∈ (Fil‘𝑌)) ∧ (𝑢 ∈ (UFil‘𝑋) ∧ (𝑋filGen𝑓) ⊆ 𝑢)) → ∃𝑔 ∈ (UFil‘𝑌)𝑓 ⊆ 𝑔)
37	14, 36	rexlimddv 3219	. . 3 ⊢ (((𝑋 ∈ UFL ∧ 𝑌 ⊆ 𝑋) ∧ 𝑓 ∈ (Fil‘𝑌)) → ∃𝑔 ∈ (UFil‘𝑌)𝑓 ⊆ 𝑔)
38	37	ralrimiva 3107	. 2 ⊢ ((𝑋 ∈ UFL ∧ 𝑌 ⊆ 𝑋) → ∀𝑓 ∈ (Fil‘𝑌)∃𝑔 ∈ (UFil‘𝑌)𝑓 ⊆ 𝑔)
39		ssexg 5242	. . . 4 ⊢ ((𝑌 ⊆ 𝑋 ∧ 𝑋 ∈ UFL) → 𝑌 ∈ V)
40	39	ancoms 458	. . 3 ⊢ ((𝑋 ∈ UFL ∧ 𝑌 ⊆ 𝑋) → 𝑌 ∈ V)
41		isufl 22972	. . 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 395   = wceq 1539   ∈ wcel 2108  ∀wral 3063  ∃wrex 3064  Vcvv 3422   ⊆ wss 3883  𝒫 cpw 4530  ‘cfv 6418  (class class class)co 7255   ↾_t crest 17048  fBascfbas 20498  filGencfg 20499  Filcfil 22904  UFilcufil 22958  UFLcufl 22959

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-1st 7804  df-2nd 7805  df-rest 17050  df-fbas 20507  df-fg 20508  df-fil 22905  df-ufil 22960  df-ufl 22961

This theorem is referenced by:  ufldom  23021