Theorem uffixfr 22982

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 22963	. . . . . . 7 ⊢ (𝐹 ∈ (UFil‘𝑋) → 𝐹 ∈ (Fil‘𝑋))
3		filtop 22914	. . . . . . 7 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝑋 ∈ 𝐹)
4	2, 3	syl 17	. . . . . 6 ⊢ (𝐹 ∈ (UFil‘𝑋) → 𝑋 ∈ 𝐹)
5		filn0 22921	. . . . . . . . 9 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐹 ≠ ∅)
6		intssuni 4898	. . . . . . . . 9 ⊢ (𝐹 ≠ ∅ → ∩ 𝐹 ⊆ ∪ 𝐹)
7	2, 5, 6	3syl 18	. . . . . . . 8 ⊢ (𝐹 ∈ (UFil‘𝑋) → ∩ 𝐹 ⊆ ∪ 𝐹)
8		filunibas 22940	. . . . . . . . 9 ⊢ (𝐹 ∈ (Fil‘𝑋) → ∪ 𝐹 = 𝑋)
9	2, 8	syl 17	. . . . . . . 8 ⊢ (𝐹 ∈ (UFil‘𝑋) → ∪ 𝐹 = 𝑋)
10	7, 9	sseqtrd 3957	. . . . . . 7 ⊢ (𝐹 ∈ (UFil‘𝑋) → ∩ 𝐹 ⊆ 𝑋)
11	10	sselda 3917	. . . . . 6 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ∈ ∩ 𝐹) → 𝐴 ∈ 𝑋)
12		uffix 22980	. . . . . 6 ⊢ ((𝑋 ∈ 𝐹 ∧ 𝐴 ∈ 𝑋) → ({{𝐴}} ∈ (fBas‘𝑋) ∧ {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥} = (𝑋filGen{{𝐴}})))
13	4, 11, 12	syl2an2r 681	. . . . 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 22937	. . . . 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 22910	. . . . 5 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐹 ⊆ 𝒫 𝑋)
21	19, 20	syl 17	. . . 4 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ∈ ∩ 𝐹) → 𝐹 ⊆ 𝒫 𝑋)
22		elintg 4884	. . . . . 6 ⊢ (𝐴 ∈ ∩ 𝐹 → (𝐴 ∈ ∩ 𝐹 ↔ ∀𝑥 ∈ 𝐹 𝐴 ∈ 𝑥))
23	22	ibi 266	. . . . 5 ⊢ (𝐴 ∈ ∩ 𝐹 → ∀𝑥 ∈ 𝐹 𝐴 ∈ 𝑥)
24	23	adantl 481	. . . 4 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ∈ ∩ 𝐹) → ∀𝑥 ∈ 𝐹 𝐴 ∈ 𝑥)
25		ssrab 4002	. . . 4 ⊢ (𝐹 ⊆ {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥} ↔ (𝐹 ⊆ 𝒫 𝑋 ∧ ∀𝑥 ∈ 𝐹 𝐴 ∈ 𝑥))
26	21, 24, 25	sylanbrc 582	. . 3 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ∈ ∩ 𝐹) → 𝐹 ⊆ {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥})
27		ufilmax 22966	. . 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 1369	. 2 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ∈ ∩ 𝐹) → 𝐹 = {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥})
29		eqimss 3973	. . . . 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 2827	. . . . . 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 579	. . . 4 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐹 = {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥}) → 𝑋 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥})
36		eleq2 2827	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝑋))
37	36	elrab 3617	. . . . 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 4884	. . . 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 256	. 2 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐹 = {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥}) → 𝐴 ∈ ∩ 𝐹)
42	28, 41	impbida 797	1 ⊢ (𝐹 ∈ (UFil‘𝑋) → (𝐴 ∈ ∩ 𝐹 ↔ 𝐹 = {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥}))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108   ≠ wne 2942  ∀wral 3063  {crab 3067   ⊆ wss 3883  ∅c0 4253  𝒫 cpw 4530  {csn 4558  ∪ cuni 4836  ∩ cint 4876  ‘cfv 6418  (class class class)co 7255  fBascfbas 20498  filGencfg 20499  Filcfil 22904  UFilcufil 22958

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-int 4877  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-fbas 20507  df-fg 20508  df-fil 22905  df-ufil 22960

This theorem is referenced by:  uffix2  22983  uffixsn  22984