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 e. ( UFil ` X ) -> ( A e. |^| F <-> F = { x e. ~P X | A e. x } ) )

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

Proof of Theorem uffixfr

Step	Hyp	Ref	Expression
1		simpl 473	. . 3 |- ( ( F e. ( UFil ` X ) /\ A e. |^| F ) -> F e. ( UFil ` X ) )
2		ufilfil 21708	. . . . . . . 8 |- ( F e. ( UFil ` X ) -> F e. ( Fil ` X ) )
3		filtop 21659	. . . . . . . 8 |- ( F e. ( Fil ` X ) -> X e. F )
4	2, 3	syl 17	. . . . . . 7 |- ( F e. ( UFil ` X ) -> X e. F )
5	4	adantr 481	. . . . . 6 |- ( ( F e. ( UFil ` X ) /\ A e. |^| F ) -> X e. F )
6		filn0 21666	. . . . . . . . 9 |- ( F e. ( Fil ` X ) -> F =/= (/) )
7		intssuni 4499	. . . . . . . . 9 |- ( F =/= (/) -> |^| F C_ U. F )
8	2, 6, 7	3syl 18	. . . . . . . 8 |- ( F e. ( UFil ` X ) -> |^| F C_ U. F )
9		filunibas 21685	. . . . . . . . 9 |- ( F e. ( Fil ` X ) -> U. F = X )
10	2, 9	syl 17	. . . . . . . 8 |- ( F e. ( UFil ` X ) -> U. F = X )
11	8, 10	sseqtrd 3641	. . . . . . 7 |- ( F e. ( UFil ` X ) -> |^| F C_ X )
12	11	sselda 3603	. . . . . 6 |- ( ( F e. ( UFil ` X ) /\ A e. |^| F ) -> A e. X )
13		uffix 21725	. . . . . 6 |- ( ( X e. F /\ A e. X ) -> ( { { A } } e. ( fBas ` X ) /\ { x e. ~P X | A e. x } = ( X filGen { { A } } ) ) )
14	5, 12, 13	syl2anc 693	. . . . 5 |- ( ( F e. ( UFil ` X ) /\ A e. |^| F ) -> ( { { A } } e. ( fBas ` X ) /\ { x e. ~P X | A e. x } = ( X filGen { { A } } ) ) )
15	14	simprd 479	. . . 4 |- ( ( F e. ( UFil ` X ) /\ A e. |^| F ) -> { x e. ~P X | A e. x } = ( X filGen { { A } } ) )
16	14	simpld 475	. . . . 5 |- ( ( F e. ( UFil ` X ) /\ A e. |^| F ) -> { { A } } e. ( fBas ` X ) )
17		fgcl 21682	. . . . 5 |- ( { { A } } e. ( fBas ` X ) -> ( X filGen { { A } } ) e. ( Fil ` X ) )
18	16, 17	syl 17	. . . 4 |- ( ( F e. ( UFil ` X ) /\ A e. |^| F ) -> ( X filGen { { A } } ) e. ( Fil ` X ) )
19	15, 18	eqeltrd 2701	. . 3 |- ( ( F e. ( UFil ` X ) /\ A e. |^| F ) -> { x e. ~P X | A e. x } e. ( Fil ` X ) )
20	2	adantr 481	. . . . 5 |- ( ( F e. ( UFil ` X ) /\ A e. |^| F ) -> F e. ( Fil ` X ) )
21		filsspw 21655	. . . . 5 |- ( F e. ( Fil ` X ) -> F C_ ~P X )
22	20, 21	syl 17	. . . 4 |- ( ( F e. ( UFil ` X ) /\ A e. |^| F ) -> F C_ ~P X )
23		elintg 4483	. . . . . 6 |- ( A e. |^| F -> ( A e. |^| F <-> A. x e. F A e. x ) )
24	23	ibi 256	. . . . 5 |- ( A e. |^| F -> A. x e. F A e. x )
25	24	adantl 482	. . . 4 |- ( ( F e. ( UFil ` X ) /\ A e. |^| F ) -> A. x e. F A e. x )
26		ssrab 3680	. . . 4 |- ( F C_ { x e. ~P X | A e. x } <-> ( F C_ ~P X /\ A. x e. F A e. x ) )
27	22, 25, 26	sylanbrc 698	. . 3 |- ( ( F e. ( UFil ` X ) /\ A e. |^| F ) -> F C_ { x e. ~P X | A e. x } )
28		ufilmax 21711	. . 3 |- ( ( F e. ( UFil ` X ) /\ { x e. ~P X | A e. x } e. ( Fil ` X ) /\ F C_ { x e. ~P X | A e. x } ) -> F = { x e. ~P X | A e. x } )
29	1, 19, 27, 28	syl3anc 1326	. 2 |- ( ( F e. ( UFil ` X ) /\ A e. |^| F ) -> F = { x e. ~P X | A e. x } )
30		eqimss 3657	. . . . 5 |- ( F = { x e. ~P X | A e. x } -> F C_ { x e. ~P X | A e. x } )
31	30	adantl 482	. . . 4 |- ( ( F e. ( UFil ` X ) /\ F = { x e. ~P X | A e. x } ) -> F C_ { x e. ~P X | A e. x } )
32	26	simprbi 480	. . . 4 |- ( F C_ { x e. ~P X | A e. x } -> A. x e. F A e. x )
33	31, 32	syl 17	. . 3 |- ( ( F e. ( UFil ` X ) /\ F = { x e. ~P X | A e. x } ) -> A. x e. F A e. x )
34		eleq2 2690	. . . . . 6 |- ( F = { x e. ~P X | A e. x } -> ( X e. F <-> X e. { x e. ~P X | A e. x } ) )
35	34	biimpac 503	. . . . 5 |- ( ( X e. F /\ F = { x e. ~P X | A e. x } ) -> X e. { x e. ~P X | A e. x } )
36	4, 35	sylan 488	. . . 4 |- ( ( F e. ( UFil ` X ) /\ F = { x e. ~P X | A e. x } ) -> X e. { x e. ~P X | A e. x } )
37		eleq2 2690	. . . . . 6 |- ( x = X -> ( A e. x <-> A e. X ) )
38	37	elrab 3363	. . . . 5 |- ( X e. { x e. ~P X | A e. x } <-> ( X e. ~P X /\ A e. X ) )
39	38	simprbi 480	. . . 4 |- ( X e. { x e. ~P X | A e. x } -> A e. X )
40		elintg 4483	. . . 4 |- ( A e. X -> ( A e. |^| F <-> A. x e. F A e. x ) )
41	36, 39, 40	3syl 18	. . 3 |- ( ( F e. ( UFil ` X ) /\ F = { x e. ~P X | A e. x } ) -> ( A e. |^| F <-> A. x e. F A e. x ) )
42	33, 41	mpbird 247	. 2 |- ( ( F e. ( UFil ` X ) /\ F = { x e. ~P X | A e. x } ) -> A e. |^| F )
43	29, 42	impbida 877	1 |- ( F e. ( UFil ` X ) -> ( A e. |^| F <-> F = { x e. ~P X | A e. x } ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   {crab 2916   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   {csn 4177   U.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