Theorem uffixfr 23947

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 23928	. . . . . . 7 ⊢ (𝐹 ∈ (UFil‘𝑋) → 𝐹 ∈ (Fil‘𝑋))
3		filtop 23879	. . . . . . 7 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝑋 ∈ 𝐹)
4	2, 3	syl 17	. . . . . 6 ⊢ (𝐹 ∈ (UFil‘𝑋) → 𝑋 ∈ 𝐹)
5		filn0 23886	. . . . . . . . 9 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐹 ≠ ∅)
6		intssuni 4975	. . . . . . . . 9 ⊢ (𝐹 ≠ ∅ → ∩ 𝐹 ⊆ ∪ 𝐹)
7	2, 5, 6	3syl 18	. . . . . . . 8 ⊢ (𝐹 ∈ (UFil‘𝑋) → ∩ 𝐹 ⊆ ∪ 𝐹)
8		filunibas 23905	. . . . . . . . 9 ⊢ (𝐹 ∈ (Fil‘𝑋) → ∪ 𝐹 = 𝑋)
9	2, 8	syl 17	. . . . . . . 8 ⊢ (𝐹 ∈ (UFil‘𝑋) → ∪ 𝐹 = 𝑋)
10	7, 9	sseqtrd 4036	. . . . . . 7 ⊢ (𝐹 ∈ (UFil‘𝑋) → ∩ 𝐹 ⊆ 𝑋)
11	10	sselda 3995	. . . . . 6 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ∈ ∩ 𝐹) → 𝐴 ∈ 𝑋)
12		uffix 23945	. . . . . 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 23902	. . . . 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 2839	. . 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 23875	. . . . 5 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐹 ⊆ 𝒫 𝑋)
21	19, 20	syl 17	. . . 4 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ∈ ∩ 𝐹) → 𝐹 ⊆ 𝒫 𝑋)
22		elintg 4959	. . . . . 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 4083	. . . 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 23931	. . 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 1370	. 2 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ∈ ∩ 𝐹) → 𝐹 = {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥})
29		eqimss 4054	. . . . 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 2828	. . . . . 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 2828	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝑋))
37	36	elrab 3695	. . . . 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 4959	. . . 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 801	1 ⊢ (𝐹 ∈ (UFil‘𝑋) → (𝐴 ∈ ∩ 𝐹 ↔ 𝐹 = {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥}))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2106   ≠ wne 2938  ∀wral 3059  {crab 3433   ⊆ wss 3963  ∅c0 4339  𝒫 cpw 4605  {csn 4631  ∪ cuni 4912  ∩ cint 4951  ‘cfv 6563  (class class class)co 7431  fBascfbas 21370  filGencfg 21371  Filcfil 23869  UFilcufil 23923

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-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438

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-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-int 4952  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-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-fbas 21379  df-fg 21380  df-fil 23870  df-ufil 23925

This theorem is referenced by:  uffix2  23948  uffixsn  23949