Theorem ssufl 23812

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 766	. . . . 5 ⊢ (((𝑋 ∈ UFL ∧ 𝑌 ⊆ 𝑋) ∧ 𝑓 ∈ (Fil‘𝑌)) → 𝑋 ∈ UFL)
2		filfbas 23742	. . . . . . . 8 ⊢ (𝑓 ∈ (Fil‘𝑌) → 𝑓 ∈ (fBas‘𝑌))
3	2	adantl 481	. . . . . . 7 ⊢ (((𝑋 ∈ UFL ∧ 𝑌 ⊆ 𝑋) ∧ 𝑓 ∈ (Fil‘𝑌)) → 𝑓 ∈ (fBas‘𝑌))
4		filsspw 23745	. . . . . . . . 9 ⊢ (𝑓 ∈ (Fil‘𝑌) → 𝑓 ⊆ 𝒫 𝑌)
5	4	adantl 481	. . . . . . . 8 ⊢ (((𝑋 ∈ UFL ∧ 𝑌 ⊆ 𝑋) ∧ 𝑓 ∈ (Fil‘𝑌)) → 𝑓 ⊆ 𝒫 𝑌)
6		simplr 768	. . . . . . . . 9 ⊢ (((𝑋 ∈ UFL ∧ 𝑌 ⊆ 𝑋) ∧ 𝑓 ∈ (Fil‘𝑌)) → 𝑌 ⊆ 𝑋)
7	6	sspwd 4579	. . . . . . . 8 ⊢ (((𝑋 ∈ UFL ∧ 𝑌 ⊆ 𝑋) ∧ 𝑓 ∈ (Fil‘𝑌)) → 𝒫 𝑌 ⊆ 𝒫 𝑋)
8	5, 7	sstrd 3960	. . . . . . 7 ⊢ (((𝑋 ∈ UFL ∧ 𝑌 ⊆ 𝑋) ∧ 𝑓 ∈ (Fil‘𝑌)) → 𝑓 ⊆ 𝒫 𝑋)
9		fbasweak 23759	. . . . . . 7 ⊢ ((𝑓 ∈ (fBas‘𝑌) ∧ 𝑓 ⊆ 𝒫 𝑋 ∧ 𝑋 ∈ UFL) → 𝑓 ∈ (fBas‘𝑋))
10	3, 8, 1, 9	syl3anc 1373	. . . . . 6 ⊢ (((𝑋 ∈ UFL ∧ 𝑌 ⊆ 𝑋) ∧ 𝑓 ∈ (Fil‘𝑌)) → 𝑓 ∈ (fBas‘𝑋))
11		fgcl 23772	. . . . . 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 23808	. . . . 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 23766	. . . . . . . . . 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 772	. . . . . . . 8 ⊢ ((((𝑋 ∈ UFL ∧ 𝑌 ⊆ 𝑋) ∧ 𝑓 ∈ (Fil‘𝑌)) ∧ (𝑢 ∈ (UFil‘𝑋) ∧ (𝑋filGen𝑓) ⊆ 𝑢)) → (𝑋filGen𝑓) ⊆ 𝑢)
19	17, 18	sstrd 3960	. . . . . . 7 ⊢ ((((𝑋 ∈ UFL ∧ 𝑌 ⊆ 𝑋) ∧ 𝑓 ∈ (Fil‘𝑌)) ∧ (𝑢 ∈ (UFil‘𝑋) ∧ (𝑋filGen𝑓) ⊆ 𝑢)) → 𝑓 ⊆ 𝑢)
20		filtop 23749	. . . . . . . 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 3950	. . . . . 6 ⊢ ((((𝑋 ∈ UFL ∧ 𝑌 ⊆ 𝑋) ∧ 𝑓 ∈ (Fil‘𝑌)) ∧ (𝑢 ∈ (UFil‘𝑋) ∧ (𝑋filGen𝑓) ⊆ 𝑢)) → 𝑌 ∈ 𝑢)
23		simprl 770	. . . . . . 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 23804	. . . . . . 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 17400	. . . . . . 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 23070	. . . . . . 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 3985	. . . . 5 ⊢ ((((𝑋 ∈ UFL ∧ 𝑌 ⊆ 𝑋) ∧ 𝑓 ∈ (Fil‘𝑌)) ∧ (𝑢 ∈ (UFil‘𝑋) ∧ (𝑋filGen𝑓) ⊆ 𝑢)) → 𝑓 ⊆ (𝑢 ↾t 𝑌))
34		sseq2 3976	. . . . . 6 ⊢ (𝑔 = (𝑢 ↾t 𝑌) → (𝑓 ⊆ 𝑔 ↔ 𝑓 ⊆ (𝑢 ↾t 𝑌)))
35	34	rspcev 3591	. . . . 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 3141	. . 3 ⊢ (((𝑋 ∈ UFL ∧ 𝑌 ⊆ 𝑋) ∧ 𝑓 ∈ (Fil‘𝑌)) → ∃𝑔 ∈ (UFil‘𝑌)𝑓 ⊆ 𝑔)
38	37	ralrimiva 3126	. 2 ⊢ ((𝑋 ∈ UFL ∧ 𝑌 ⊆ 𝑋) → ∀𝑓 ∈ (Fil‘𝑌)∃𝑔 ∈ (UFil‘𝑌)𝑓 ⊆ 𝑔)
39		ssexg 5281	. . . 4 ⊢ ((𝑌 ⊆ 𝑋 ∧ 𝑋 ∈ UFL) → 𝑌 ∈ V)
40	39	ancoms 458	. . 3 ⊢ ((𝑋 ∈ UFL ∧ 𝑌 ⊆ 𝑋) → 𝑌 ∈ V)
41		isufl 23807	. . 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 1540   ∈ wcel 2109  ∀wral 3045  ∃wrex 3054  Vcvv 3450   ⊆ wss 3917  𝒫 cpw 4566  ‘cfv 6514  (class class class)co 7390   ↾_t crest 17390  fBascfbas 21259  filGencfg 21260  Filcfil 23739  UFilcufil 23793  UFLcufl 23794

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-ov 7393  df-oprab 7394  df-mpo 7395  df-1st 7971  df-2nd 7972  df-rest 17392  df-fbas 21268  df-fg 21269  df-fil 23740  df-ufil 23795  df-ufl 23796

This theorem is referenced by:  ufldom  23856