Theorem ssufl 21943

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 744	. . . . 5 ⊢ (((𝑋 ∈ UFL ∧ 𝑌 ⊆ 𝑋) ∧ 𝑓 ∈ (Fil‘𝑌)) → 𝑋 ∈ UFL)
2		filfbas 21873	. . . . . . . 8 ⊢ (𝑓 ∈ (Fil‘𝑌) → 𝑓 ∈ (fBas‘𝑌))
3	2	adantl 467	. . . . . . 7 ⊢ (((𝑋 ∈ UFL ∧ 𝑌 ⊆ 𝑋) ∧ 𝑓 ∈ (Fil‘𝑌)) → 𝑓 ∈ (fBas‘𝑌))
4		filsspw 21876	. . . . . . . . 9 ⊢ (𝑓 ∈ (Fil‘𝑌) → 𝑓 ⊆ 𝒫 𝑌)
5	4	adantl 467	. . . . . . . 8 ⊢ (((𝑋 ∈ UFL ∧ 𝑌 ⊆ 𝑋) ∧ 𝑓 ∈ (Fil‘𝑌)) → 𝑓 ⊆ 𝒫 𝑌)
6		simplr 746	. . . . . . . . 9 ⊢ (((𝑋 ∈ UFL ∧ 𝑌 ⊆ 𝑋) ∧ 𝑓 ∈ (Fil‘𝑌)) → 𝑌 ⊆ 𝑋)
7		sspwb 5046	. . . . . . . . 9 ⊢ (𝑌 ⊆ 𝑋 ↔ 𝒫 𝑌 ⊆ 𝒫 𝑋)
8	6, 7	sylib 208	. . . . . . . 8 ⊢ (((𝑋 ∈ UFL ∧ 𝑌 ⊆ 𝑋) ∧ 𝑓 ∈ (Fil‘𝑌)) → 𝒫 𝑌 ⊆ 𝒫 𝑋)
9	5, 8	sstrd 3763	. . . . . . 7 ⊢ (((𝑋 ∈ UFL ∧ 𝑌 ⊆ 𝑋) ∧ 𝑓 ∈ (Fil‘𝑌)) → 𝑓 ⊆ 𝒫 𝑋)
10		fbasweak 21890	. . . . . . 7 ⊢ ((𝑓 ∈ (fBas‘𝑌) ∧ 𝑓 ⊆ 𝒫 𝑋 ∧ 𝑋 ∈ UFL) → 𝑓 ∈ (fBas‘𝑋))
11	3, 9, 1, 10	syl3anc 1476	. . . . . 6 ⊢ (((𝑋 ∈ UFL ∧ 𝑌 ⊆ 𝑋) ∧ 𝑓 ∈ (Fil‘𝑌)) → 𝑓 ∈ (fBas‘𝑋))
12		fgcl 21903	. . . . . 6 ⊢ (𝑓 ∈ (fBas‘𝑋) → (𝑋filGen𝑓) ∈ (Fil‘𝑋))
13	11, 12	syl 17	. . . . 5 ⊢ (((𝑋 ∈ UFL ∧ 𝑌 ⊆ 𝑋) ∧ 𝑓 ∈ (Fil‘𝑌)) → (𝑋filGen𝑓) ∈ (Fil‘𝑋))
14		ufli 21939	. . . . 5 ⊢ ((𝑋 ∈ UFL ∧ (𝑋filGen𝑓) ∈ (Fil‘𝑋)) → ∃𝑢 ∈ (UFil‘𝑋)(𝑋filGen𝑓) ⊆ 𝑢)
15	1, 13, 14	syl2anc 567	. . . 4 ⊢ (((𝑋 ∈ UFL ∧ 𝑌 ⊆ 𝑋) ∧ 𝑓 ∈ (Fil‘𝑌)) → ∃𝑢 ∈ (UFil‘𝑋)(𝑋filGen𝑓) ⊆ 𝑢)
16		ssfg 21897	. . . . . . . . . 10 ⊢ (𝑓 ∈ (fBas‘𝑋) → 𝑓 ⊆ (𝑋filGen𝑓))
17	11, 16	syl 17	. . . . . . . . 9 ⊢ (((𝑋 ∈ UFL ∧ 𝑌 ⊆ 𝑋) ∧ 𝑓 ∈ (Fil‘𝑌)) → 𝑓 ⊆ (𝑋filGen𝑓))
18	17	adantr 466	. . . . . . . 8 ⊢ ((((𝑋 ∈ UFL ∧ 𝑌 ⊆ 𝑋) ∧ 𝑓 ∈ (Fil‘𝑌)) ∧ (𝑢 ∈ (UFil‘𝑋) ∧ (𝑋filGen𝑓) ⊆ 𝑢)) → 𝑓 ⊆ (𝑋filGen𝑓))
19		simprr 750	. . . . . . . 8 ⊢ ((((𝑋 ∈ UFL ∧ 𝑌 ⊆ 𝑋) ∧ 𝑓 ∈ (Fil‘𝑌)) ∧ (𝑢 ∈ (UFil‘𝑋) ∧ (𝑋filGen𝑓) ⊆ 𝑢)) → (𝑋filGen𝑓) ⊆ 𝑢)
20	18, 19	sstrd 3763	. . . . . . 7 ⊢ ((((𝑋 ∈ UFL ∧ 𝑌 ⊆ 𝑋) ∧ 𝑓 ∈ (Fil‘𝑌)) ∧ (𝑢 ∈ (UFil‘𝑋) ∧ (𝑋filGen𝑓) ⊆ 𝑢)) → 𝑓 ⊆ 𝑢)
21		filtop 21880	. . . . . . . 8 ⊢ (𝑓 ∈ (Fil‘𝑌) → 𝑌 ∈ 𝑓)
22	21	ad2antlr 700	. . . . . . 7 ⊢ ((((𝑋 ∈ UFL ∧ 𝑌 ⊆ 𝑋) ∧ 𝑓 ∈ (Fil‘𝑌)) ∧ (𝑢 ∈ (UFil‘𝑋) ∧ (𝑋filGen𝑓) ⊆ 𝑢)) → 𝑌 ∈ 𝑓)
23	20, 22	sseldd 3754	. . . . . 6 ⊢ ((((𝑋 ∈ UFL ∧ 𝑌 ⊆ 𝑋) ∧ 𝑓 ∈ (Fil‘𝑌)) ∧ (𝑢 ∈ (UFil‘𝑋) ∧ (𝑋filGen𝑓) ⊆ 𝑢)) → 𝑌 ∈ 𝑢)
24		simprl 748	. . . . . . 7 ⊢ ((((𝑋 ∈ UFL ∧ 𝑌 ⊆ 𝑋) ∧ 𝑓 ∈ (Fil‘𝑌)) ∧ (𝑢 ∈ (UFil‘𝑋) ∧ (𝑋filGen𝑓) ⊆ 𝑢)) → 𝑢 ∈ (UFil‘𝑋))
25	6	adantr 466	. . . . . . 7 ⊢ ((((𝑋 ∈ UFL ∧ 𝑌 ⊆ 𝑋) ∧ 𝑓 ∈ (Fil‘𝑌)) ∧ (𝑢 ∈ (UFil‘𝑋) ∧ (𝑋filGen𝑓) ⊆ 𝑢)) → 𝑌 ⊆ 𝑋)
26		trufil 21935	. . . . . . 7 ⊢ ((𝑢 ∈ (UFil‘𝑋) ∧ 𝑌 ⊆ 𝑋) → ((𝑢 ↾t 𝑌) ∈ (UFil‘𝑌) ↔ 𝑌 ∈ 𝑢))
27	24, 25, 26	syl2anc 567	. . . . . 6 ⊢ ((((𝑋 ∈ UFL ∧ 𝑌 ⊆ 𝑋) ∧ 𝑓 ∈ (Fil‘𝑌)) ∧ (𝑢 ∈ (UFil‘𝑋) ∧ (𝑋filGen𝑓) ⊆ 𝑢)) → ((𝑢 ↾t 𝑌) ∈ (UFil‘𝑌) ↔ 𝑌 ∈ 𝑢))
28	23, 27	mpbird 247	. . . . 5 ⊢ ((((𝑋 ∈ UFL ∧ 𝑌 ⊆ 𝑋) ∧ 𝑓 ∈ (Fil‘𝑌)) ∧ (𝑢 ∈ (UFil‘𝑋) ∧ (𝑋filGen𝑓) ⊆ 𝑢)) → (𝑢 ↾t 𝑌) ∈ (UFil‘𝑌))
29	5	adantr 466	. . . . . . 7 ⊢ ((((𝑋 ∈ UFL ∧ 𝑌 ⊆ 𝑋) ∧ 𝑓 ∈ (Fil‘𝑌)) ∧ (𝑢 ∈ (UFil‘𝑋) ∧ (𝑋filGen𝑓) ⊆ 𝑢)) → 𝑓 ⊆ 𝒫 𝑌)
30		restid2 16300	. . . . . . 7 ⊢ ((𝑌 ∈ 𝑓 ∧ 𝑓 ⊆ 𝒫 𝑌) → (𝑓 ↾t 𝑌) = 𝑓)
31	22, 29, 30	syl2anc 567	. . . . . 6 ⊢ ((((𝑋 ∈ UFL ∧ 𝑌 ⊆ 𝑋) ∧ 𝑓 ∈ (Fil‘𝑌)) ∧ (𝑢 ∈ (UFil‘𝑋) ∧ (𝑋filGen𝑓) ⊆ 𝑢)) → (𝑓 ↾t 𝑌) = 𝑓)
32		ssrest 21202	. . . . . . 7 ⊢ ((𝑢 ∈ (UFil‘𝑋) ∧ 𝑓 ⊆ 𝑢) → (𝑓 ↾t 𝑌) ⊆ (𝑢 ↾t 𝑌))
33	24, 20, 32	syl2anc 567	. . . . . 6 ⊢ ((((𝑋 ∈ UFL ∧ 𝑌 ⊆ 𝑋) ∧ 𝑓 ∈ (Fil‘𝑌)) ∧ (𝑢 ∈ (UFil‘𝑋) ∧ (𝑋filGen𝑓) ⊆ 𝑢)) → (𝑓 ↾t 𝑌) ⊆ (𝑢 ↾t 𝑌))
34	31, 33	eqsstr3d 3790	. . . . 5 ⊢ ((((𝑋 ∈ UFL ∧ 𝑌 ⊆ 𝑋) ∧ 𝑓 ∈ (Fil‘𝑌)) ∧ (𝑢 ∈ (UFil‘𝑋) ∧ (𝑋filGen𝑓) ⊆ 𝑢)) → 𝑓 ⊆ (𝑢 ↾t 𝑌))
35		sseq2 3777	. . . . . 6 ⊢ (𝑔 = (𝑢 ↾t 𝑌) → (𝑓 ⊆ 𝑔 ↔ 𝑓 ⊆ (𝑢 ↾t 𝑌)))
36	35	rspcev 3461	. . . . 5 ⊢ (((𝑢 ↾t 𝑌) ∈ (UFil‘𝑌) ∧ 𝑓 ⊆ (𝑢 ↾t 𝑌)) → ∃𝑔 ∈ (UFil‘𝑌)𝑓 ⊆ 𝑔)
37	28, 34, 36	syl2anc 567	. . . 4 ⊢ ((((𝑋 ∈ UFL ∧ 𝑌 ⊆ 𝑋) ∧ 𝑓 ∈ (Fil‘𝑌)) ∧ (𝑢 ∈ (UFil‘𝑋) ∧ (𝑋filGen𝑓) ⊆ 𝑢)) → ∃𝑔 ∈ (UFil‘𝑌)𝑓 ⊆ 𝑔)
38	15, 37	rexlimddv 3183	. . 3 ⊢ (((𝑋 ∈ UFL ∧ 𝑌 ⊆ 𝑋) ∧ 𝑓 ∈ (Fil‘𝑌)) → ∃𝑔 ∈ (UFil‘𝑌)𝑓 ⊆ 𝑔)
39	38	ralrimiva 3115	. 2 ⊢ ((𝑋 ∈ UFL ∧ 𝑌 ⊆ 𝑋) → ∀𝑓 ∈ (Fil‘𝑌)∃𝑔 ∈ (UFil‘𝑌)𝑓 ⊆ 𝑔)
40		ssexg 4939	. . . 4 ⊢ ((𝑌 ⊆ 𝑋 ∧ 𝑋 ∈ UFL) → 𝑌 ∈ V)
41	40	ancoms 455	. . 3 ⊢ ((𝑋 ∈ UFL ∧ 𝑌 ⊆ 𝑋) → 𝑌 ∈ V)
42		isufl 21938	. . 3 ⊢ (𝑌 ∈ V → (𝑌 ∈ UFL ↔ ∀𝑓 ∈ (Fil‘𝑌)∃𝑔 ∈ (UFil‘𝑌)𝑓 ⊆ 𝑔))
43	41, 42	syl 17	. 2 ⊢ ((𝑋 ∈ UFL ∧ 𝑌 ⊆ 𝑋) → (𝑌 ∈ UFL ↔ ∀𝑓 ∈ (Fil‘𝑌)∃𝑔 ∈ (UFil‘𝑌)𝑓 ⊆ 𝑔))
44	39, 43	mpbird 247	1 ⊢ ((𝑋 ∈ UFL ∧ 𝑌 ⊆ 𝑋) → 𝑌 ∈ UFL)

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 382   = wceq 1631   ∈ wcel 2145  ∀wral 3061  ∃wrex 3062  Vcvv 3351   ⊆ wss 3724  𝒫 cpw 4298  ‘cfv 6032  (class class class)co 6794   ↾_t crest 16290  fBascfbas 19950  filGencfg 19951  Filcfil 21870  UFilcufil 21924  UFLcufl 21925

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7097

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 829  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3589  df-csb 3684  df-dif 3727  df-un 3729  df-in 3731  df-ss 3738  df-nul 4065  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5995  df-fun 6034  df-fn 6035  df-f 6036  df-f1 6037  df-fo 6038  df-f1o 6039  df-fv 6040  df-ov 6797  df-oprab 6798  df-mpt2 6799  df-1st 7316  df-2nd 7317  df-rest 16292  df-fbas 19959  df-fg 19960  df-fil 21871  df-ufil 21926  df-ufl 21927

This theorem is referenced by:  ufldom  21987