Theorem uffixfr 21727

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 473	. . 3 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ∈ ∩ 𝐹) → 𝐹 ∈ (UFil‘𝑋))
2		ufilfil 21708	. . . . . . . 8 ⊢ (𝐹 ∈ (UFil‘𝑋) → 𝐹 ∈ (Fil‘𝑋))
3		filtop 21659	. . . . . . . 8 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝑋 ∈ 𝐹)
4	2, 3	syl 17	. . . . . . 7 ⊢ (𝐹 ∈ (UFil‘𝑋) → 𝑋 ∈ 𝐹)
5	4	adantr 481	. . . . . 6 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ∈ ∩ 𝐹) → 𝑋 ∈ 𝐹)
6		filn0 21666	. . . . . . . . 9 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐹 ≠ ∅)
7		intssuni 4499	. . . . . . . . 9 ⊢ (𝐹 ≠ ∅ → ∩ 𝐹 ⊆ ∪ 𝐹)
8	2, 6, 7	3syl 18	. . . . . . . 8 ⊢ (𝐹 ∈ (UFil‘𝑋) → ∩ 𝐹 ⊆ ∪ 𝐹)
9		filunibas 21685	. . . . . . . . 9 ⊢ (𝐹 ∈ (Fil‘𝑋) → ∪ 𝐹 = 𝑋)
10	2, 9	syl 17	. . . . . . . 8 ⊢ (𝐹 ∈ (UFil‘𝑋) → ∪ 𝐹 = 𝑋)
11	8, 10	sseqtrd 3641	. . . . . . 7 ⊢ (𝐹 ∈ (UFil‘𝑋) → ∩ 𝐹 ⊆ 𝑋)
12	11	sselda 3603	. . . . . 6 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ∈ ∩ 𝐹) → 𝐴 ∈ 𝑋)
13		uffix 21725	. . . . . 6 ⊢ ((𝑋 ∈ 𝐹 ∧ 𝐴 ∈ 𝑋) → ({{𝐴}} ∈ (fBas‘𝑋) ∧ {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥} = (𝑋filGen{{𝐴}})))
14	5, 12, 13	syl2anc 693	. . . . 5 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ∈ ∩ 𝐹) → ({{𝐴}} ∈ (fBas‘𝑋) ∧ {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥} = (𝑋filGen{{𝐴}})))
15	14	simprd 479	. . . 4 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ∈ ∩ 𝐹) → {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥} = (𝑋filGen{{𝐴}}))
16	14	simpld 475	. . . . 5 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ∈ ∩ 𝐹) → {{𝐴}} ∈ (fBas‘𝑋))
17		fgcl 21682	. . . . 5 ⊢ ({{𝐴}} ∈ (fBas‘𝑋) → (𝑋filGen{{𝐴}}) ∈ (Fil‘𝑋))
18	16, 17	syl 17	. . . 4 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ∈ ∩ 𝐹) → (𝑋filGen{{𝐴}}) ∈ (Fil‘𝑋))
19	15, 18	eqeltrd 2701	. . 3 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ∈ ∩ 𝐹) → {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥} ∈ (Fil‘𝑋))
20	2	adantr 481	. . . . 5 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ∈ ∩ 𝐹) → 𝐹 ∈ (Fil‘𝑋))
21		filsspw 21655	. . . . 5 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐹 ⊆ 𝒫 𝑋)
22	20, 21	syl 17	. . . 4 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ∈ ∩ 𝐹) → 𝐹 ⊆ 𝒫 𝑋)
23		elintg 4483	. . . . . 6 ⊢ (𝐴 ∈ ∩ 𝐹 → (𝐴 ∈ ∩ 𝐹 ↔ ∀𝑥 ∈ 𝐹 𝐴 ∈ 𝑥))
24	23	ibi 256	. . . . 5 ⊢ (𝐴 ∈ ∩ 𝐹 → ∀𝑥 ∈ 𝐹 𝐴 ∈ 𝑥)
25	24	adantl 482	. . . 4 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ∈ ∩ 𝐹) → ∀𝑥 ∈ 𝐹 𝐴 ∈ 𝑥)
26		ssrab 3680	. . . 4 ⊢ (𝐹 ⊆ {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥} ↔ (𝐹 ⊆ 𝒫 𝑋 ∧ ∀𝑥 ∈ 𝐹 𝐴 ∈ 𝑥))
27	22, 25, 26	sylanbrc 698	. . 3 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ∈ ∩ 𝐹) → 𝐹 ⊆ {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥})
28		ufilmax 21711	. . 3 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥} ∈ (Fil‘𝑋) ∧ 𝐹 ⊆ {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥}) → 𝐹 = {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥})
29	1, 19, 27, 28	syl3anc 1326	. 2 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ∈ ∩ 𝐹) → 𝐹 = {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥})
30		eqimss 3657	. . . . 5 ⊢ (𝐹 = {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥} → 𝐹 ⊆ {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥})
31	30	adantl 482	. . . 4 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐹 = {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥}) → 𝐹 ⊆ {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥})
32	26	simprbi 480	. . . 4 ⊢ (𝐹 ⊆ {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥} → ∀𝑥 ∈ 𝐹 𝐴 ∈ 𝑥)
33	31, 32	syl 17	. . 3 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐹 = {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥}) → ∀𝑥 ∈ 𝐹 𝐴 ∈ 𝑥)
34		eleq2 2690	. . . . . 6 ⊢ (𝐹 = {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥} → (𝑋 ∈ 𝐹 ↔ 𝑋 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥}))
35	34	biimpac 503	. . . . 5 ⊢ ((𝑋 ∈ 𝐹 ∧ 𝐹 = {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥}) → 𝑋 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥})
36	4, 35	sylan 488	. . . 4 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐹 = {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥}) → 𝑋 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥})
37		eleq2 2690	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝑋))
38	37	elrab 3363	. . . . 5 ⊢ (𝑋 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥} ↔ (𝑋 ∈ 𝒫 𝑋 ∧ 𝐴 ∈ 𝑋))
39	38	simprbi 480	. . . 4 ⊢ (𝑋 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥} → 𝐴 ∈ 𝑋)
40		elintg 4483	. . . 4 ⊢ (𝐴 ∈ 𝑋 → (𝐴 ∈ ∩ 𝐹 ↔ ∀𝑥 ∈ 𝐹 𝐴 ∈ 𝑥))
41	36, 39, 40	3syl 18	. . 3 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐹 = {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥}) → (𝐴 ∈ ∩ 𝐹 ↔ ∀𝑥 ∈ 𝐹 𝐴 ∈ 𝑥))
42	33, 41	mpbird 247	. 2 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐹 = {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥}) → 𝐴 ∈ ∩ 𝐹)
43	29, 42	impbida 877	1 ⊢ (𝐹 ∈ (UFil‘𝑋) → (𝐴 ∈ ∩ 𝐹 ↔ 𝐹 = {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥}))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990   ≠ wne 2794  ∀wral 2912  {crab 2916   ⊆ wss 3574  ∅c0 3915  𝒫 cpw 4158  {csn 4177  ∪ cuni 4436  ∩ cint 4475  ‘cfv 5888  (class class class)co 6650  fBascfbas 19734  filGencfg 19735  Filcfil 21649  UFilcufil 21703

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-int 4476  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-fbas 19743  df-fg 19744  df-fil 21650  df-ufil 21705

This theorem is referenced by:  uffix2  21728  uffixsn  21729