Theorem ssufl 23942

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 767	. . . . 5 ⊢ (((𝑋 ∈ UFL ∧ 𝑌 ⊆ 𝑋) ∧ 𝑓 ∈ (Fil‘𝑌)) → 𝑋 ∈ UFL)
2		filfbas 23872	. . . . . . . 8 ⊢ (𝑓 ∈ (Fil‘𝑌) → 𝑓 ∈ (fBas‘𝑌))
3	2	adantl 481	. . . . . . 7 ⊢ (((𝑋 ∈ UFL ∧ 𝑌 ⊆ 𝑋) ∧ 𝑓 ∈ (Fil‘𝑌)) → 𝑓 ∈ (fBas‘𝑌))
4		filsspw 23875	. . . . . . . . 9 ⊢ (𝑓 ∈ (Fil‘𝑌) → 𝑓 ⊆ 𝒫 𝑌)
5	4	adantl 481	. . . . . . . 8 ⊢ (((𝑋 ∈ UFL ∧ 𝑌 ⊆ 𝑋) ∧ 𝑓 ∈ (Fil‘𝑌)) → 𝑓 ⊆ 𝒫 𝑌)
6		simplr 769	. . . . . . . . 9 ⊢ (((𝑋 ∈ UFL ∧ 𝑌 ⊆ 𝑋) ∧ 𝑓 ∈ (Fil‘𝑌)) → 𝑌 ⊆ 𝑋)
7	6	sspwd 4618	. . . . . . . 8 ⊢ (((𝑋 ∈ UFL ∧ 𝑌 ⊆ 𝑋) ∧ 𝑓 ∈ (Fil‘𝑌)) → 𝒫 𝑌 ⊆ 𝒫 𝑋)
8	5, 7	sstrd 4006	. . . . . . 7 ⊢ (((𝑋 ∈ UFL ∧ 𝑌 ⊆ 𝑋) ∧ 𝑓 ∈ (Fil‘𝑌)) → 𝑓 ⊆ 𝒫 𝑋)
9		fbasweak 23889	. . . . . . 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 23902	. . . . . 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 23938	. . . . 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 23896	. . . . . . . . . 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 773	. . . . . . . 8 ⊢ ((((𝑋 ∈ UFL ∧ 𝑌 ⊆ 𝑋) ∧ 𝑓 ∈ (Fil‘𝑌)) ∧ (𝑢 ∈ (UFil‘𝑋) ∧ (𝑋filGen𝑓) ⊆ 𝑢)) → (𝑋filGen𝑓) ⊆ 𝑢)
19	17, 18	sstrd 4006	. . . . . . 7 ⊢ ((((𝑋 ∈ UFL ∧ 𝑌 ⊆ 𝑋) ∧ 𝑓 ∈ (Fil‘𝑌)) ∧ (𝑢 ∈ (UFil‘𝑋) ∧ (𝑋filGen𝑓) ⊆ 𝑢)) → 𝑓 ⊆ 𝑢)
20		filtop 23879	. . . . . . . 8 ⊢ (𝑓 ∈ (Fil‘𝑌) → 𝑌 ∈ 𝑓)
21	20	ad2antlr 727	. . . . . . 7 ⊢ ((((𝑋 ∈ UFL ∧ 𝑌 ⊆ 𝑋) ∧ 𝑓 ∈ (Fil‘𝑌)) ∧ (𝑢 ∈ (UFil‘𝑋) ∧ (𝑋filGen𝑓) ⊆ 𝑢)) → 𝑌 ∈ 𝑓)
22	19, 21	sseldd 3996	. . . . . 6 ⊢ ((((𝑋 ∈ UFL ∧ 𝑌 ⊆ 𝑋) ∧ 𝑓 ∈ (Fil‘𝑌)) ∧ (𝑢 ∈ (UFil‘𝑋) ∧ (𝑋filGen𝑓) ⊆ 𝑢)) → 𝑌 ∈ 𝑢)
23		simprl 771	. . . . . . 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 23934	. . . . . . 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 257	. . . . 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 17477	. . . . . . 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 23200	. . . . . . 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 4035	. . . . 5 ⊢ ((((𝑋 ∈ UFL ∧ 𝑌 ⊆ 𝑋) ∧ 𝑓 ∈ (Fil‘𝑌)) ∧ (𝑢 ∈ (UFil‘𝑋) ∧ (𝑋filGen𝑓) ⊆ 𝑢)) → 𝑓 ⊆ (𝑢 ↾t 𝑌))
34		sseq2 4022	. . . . . 6 ⊢ (𝑔 = (𝑢 ↾t 𝑌) → (𝑓 ⊆ 𝑔 ↔ 𝑓 ⊆ (𝑢 ↾t 𝑌)))
35	34	rspcev 3622	. . . . 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 3159	. . 3 ⊢ (((𝑋 ∈ UFL ∧ 𝑌 ⊆ 𝑋) ∧ 𝑓 ∈ (Fil‘𝑌)) → ∃𝑔 ∈ (UFil‘𝑌)𝑓 ⊆ 𝑔)
38	37	ralrimiva 3144	. 2 ⊢ ((𝑋 ∈ UFL ∧ 𝑌 ⊆ 𝑋) → ∀𝑓 ∈ (Fil‘𝑌)∃𝑔 ∈ (UFil‘𝑌)𝑓 ⊆ 𝑔)
39		ssexg 5329	. . . 4 ⊢ ((𝑌 ⊆ 𝑋 ∧ 𝑋 ∈ UFL) → 𝑌 ∈ V)
40	39	ancoms 458	. . 3 ⊢ ((𝑋 ∈ UFL ∧ 𝑌 ⊆ 𝑋) → 𝑌 ∈ V)
41		isufl 23937	. . 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 257	1 ⊢ ((𝑋 ∈ UFL ∧ 𝑌 ⊆ 𝑋) → 𝑌 ∈ UFL)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2106  ∀wral 3059  ∃wrex 3068  Vcvv 3478   ⊆ wss 3963  𝒫 cpw 4605  ‘cfv 6563  (class class class)co 7431   ↾_t crest 17467  fBascfbas 21370  filGencfg 21371  Filcfil 23869  UFilcufil 23923  UFLcufl 23924

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-rest 17469  df-fbas 21379  df-fg 21380  df-fil 23870  df-ufil 23925  df-ufl 23926

This theorem is referenced by:  ufldom  23986