Theorem uffixfr 23808

Description: An ultrafilter is either fixed or free. A fixed ultrafilter is called principal (generated by a single element 𝐴), and a free ultrafilter is called nonprincipal (having empty intersection). Note that examples of free ultrafilters cannot be defined in ZFC without some form of global choice. (Contributed by Jeff Hankins, 4-Dec-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)

Assertion

Ref	Expression
uffixfr	⊢ (𝐹 ∈ (UFil‘𝑋) → (𝐴 ∈ ∩ 𝐹 ↔ 𝐹 = {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥}))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐹   𝑥,𝑋

Proof of Theorem uffixfr

Step	Hyp	Ref	Expression
1		simpl 482	. . 3 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ∈ ∩ 𝐹) → 𝐹 ∈ (UFil‘𝑋))
2		ufilfil 23789	. . . . . . 7 ⊢ (𝐹 ∈ (UFil‘𝑋) → 𝐹 ∈ (Fil‘𝑋))
3		filtop 23740	. . . . . . 7 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝑋 ∈ 𝐹)
4	2, 3	syl 17	. . . . . 6 ⊢ (𝐹 ∈ (UFil‘𝑋) → 𝑋 ∈ 𝐹)
5		filn0 23747	. . . . . . . . 9 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐹 ≠ ∅)
6		intssuni 4920	. . . . . . . . 9 ⊢ (𝐹 ≠ ∅ → ∩ 𝐹 ⊆ ∪ 𝐹)
7	2, 5, 6	3syl 18	. . . . . . . 8 ⊢ (𝐹 ∈ (UFil‘𝑋) → ∩ 𝐹 ⊆ ∪ 𝐹)
8		filunibas 23766	. . . . . . . . 9 ⊢ (𝐹 ∈ (Fil‘𝑋) → ∪ 𝐹 = 𝑋)
9	2, 8	syl 17	. . . . . . . 8 ⊢ (𝐹 ∈ (UFil‘𝑋) → ∪ 𝐹 = 𝑋)
10	7, 9	sseqtrd 3972	. . . . . . 7 ⊢ (𝐹 ∈ (UFil‘𝑋) → ∩ 𝐹 ⊆ 𝑋)
11	10	sselda 3935	. . . . . 6 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ∈ ∩ 𝐹) → 𝐴 ∈ 𝑋)
12		uffix 23806	. . . . . 6 ⊢ ((𝑋 ∈ 𝐹 ∧ 𝐴 ∈ 𝑋) → ({{𝐴}} ∈ (fBas‘𝑋) ∧ {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥} = (𝑋filGen{{𝐴}})))
13	4, 11, 12	syl2an2r 685	. . . . 5 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ∈ ∩ 𝐹) → ({{𝐴}} ∈ (fBas‘𝑋) ∧ {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥} = (𝑋filGen{{𝐴}})))
14	13	simprd 495	. . . 4 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ∈ ∩ 𝐹) → {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥} = (𝑋filGen{{𝐴}}))
15	13	simpld 494	. . . . 5 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ∈ ∩ 𝐹) → {{𝐴}} ∈ (fBas‘𝑋))
16		fgcl 23763	. . . . 5 ⊢ ({{𝐴}} ∈ (fBas‘𝑋) → (𝑋filGen{{𝐴}}) ∈ (Fil‘𝑋))
17	15, 16	syl 17	. . . 4 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ∈ ∩ 𝐹) → (𝑋filGen{{𝐴}}) ∈ (Fil‘𝑋))
18	14, 17	eqeltrd 2828	. . 3 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ∈ ∩ 𝐹) → {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥} ∈ (Fil‘𝑋))
19	2	adantr 480	. . . . 5 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ∈ ∩ 𝐹) → 𝐹 ∈ (Fil‘𝑋))
20		filsspw 23736	. . . . 5 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐹 ⊆ 𝒫 𝑋)
21	19, 20	syl 17	. . . 4 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ∈ ∩ 𝐹) → 𝐹 ⊆ 𝒫 𝑋)
22		elintg 4904	. . . . . 6 ⊢ (𝐴 ∈ ∩ 𝐹 → (𝐴 ∈ ∩ 𝐹 ↔ ∀𝑥 ∈ 𝐹 𝐴 ∈ 𝑥))
23	22	ibi 267	. . . . 5 ⊢ (𝐴 ∈ ∩ 𝐹 → ∀𝑥 ∈ 𝐹 𝐴 ∈ 𝑥)
24	23	adantl 481	. . . 4 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ∈ ∩ 𝐹) → ∀𝑥 ∈ 𝐹 𝐴 ∈ 𝑥)
25		ssrab 4024	. . . 4 ⊢ (𝐹 ⊆ {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥} ↔ (𝐹 ⊆ 𝒫 𝑋 ∧ ∀𝑥 ∈ 𝐹 𝐴 ∈ 𝑥))
26	21, 24, 25	sylanbrc 583	. . 3 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ∈ ∩ 𝐹) → 𝐹 ⊆ {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥})
27		ufilmax 23792	. . 3 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥} ∈ (Fil‘𝑋) ∧ 𝐹 ⊆ {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥}) → 𝐹 = {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥})
28	1, 18, 26, 27	syl3anc 1373	. 2 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ∈ ∩ 𝐹) → 𝐹 = {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥})
29		eqimss 3994	. . . . 5 ⊢ (𝐹 = {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥} → 𝐹 ⊆ {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥})
30	29	adantl 481	. . . 4 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐹 = {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥}) → 𝐹 ⊆ {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥})
31	25	simprbi 496	. . . 4 ⊢ (𝐹 ⊆ {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥} → ∀𝑥 ∈ 𝐹 𝐴 ∈ 𝑥)
32	30, 31	syl 17	. . 3 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐹 = {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥}) → ∀𝑥 ∈ 𝐹 𝐴 ∈ 𝑥)
33		eleq2 2817	. . . . . 6 ⊢ (𝐹 = {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥} → (𝑋 ∈ 𝐹 ↔ 𝑋 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥}))
34	33	biimpac 478	. . . . 5 ⊢ ((𝑋 ∈ 𝐹 ∧ 𝐹 = {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥}) → 𝑋 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥})
35	4, 34	sylan 580	. . . 4 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐹 = {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥}) → 𝑋 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥})
36		eleq2 2817	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝑋))
37	36	elrab 3648	. . . . 5 ⊢ (𝑋 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥} ↔ (𝑋 ∈ 𝒫 𝑋 ∧ 𝐴 ∈ 𝑋))
38	37	simprbi 496	. . . 4 ⊢ (𝑋 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥} → 𝐴 ∈ 𝑋)
39		elintg 4904	. . . 4 ⊢ (𝐴 ∈ 𝑋 → (𝐴 ∈ ∩ 𝐹 ↔ ∀𝑥 ∈ 𝐹 𝐴 ∈ 𝑥))
40	35, 38, 39	3syl 18	. . 3 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐹 = {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥}) → (𝐴 ∈ ∩ 𝐹 ↔ ∀𝑥 ∈ 𝐹 𝐴 ∈ 𝑥))
41	32, 40	mpbird 257	. 2 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐹 = {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥}) → 𝐴 ∈ ∩ 𝐹)
42	28, 41	impbida 800	1 ⊢ (𝐹 ∈ (UFil‘𝑋) → (𝐴 ∈ ∩ 𝐹 ↔ 𝐹 = {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥}))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  ∀wral 3044  {crab 3394   ⊆ wss 3903  ∅c0 4284  𝒫 cpw 4551  {csn 4577  ∪ cuni 4858  ∩ cint 4896  ‘cfv 6482  (class class class)co 7349  fBascfbas 21249  filGencfg 21250  Filcfil 23730  UFilcufil 23784

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 2701  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-int 4897  df-br 5093  df-opab 5155  df-mpt 5174  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6438  df-fun 6484  df-fv 6490  df-ov 7352  df-oprab 7353  df-mpo 7354  df-fbas 21258  df-fg 21259  df-fil 23731  df-ufil 23786

This theorem is referenced by:  uffix2  23809  uffixsn  23810