Theorem zfbas 21700

Description: The set of upper sets of integers is a filter base on ZZ, which corresponds to convergence of sequences on ZZ. (Contributed by Mario Carneiro, 13-Oct-2015.)

Assertion

Ref	Expression
zfbas	|- ran ZZ>= e. ( fBas ` ZZ )

Proof of Theorem zfbas

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		uzf 11690	. . 3 |- ZZ>= : ZZ --> ~P ZZ
2		frn 6053	. . 3 |- ( ZZ>= : ZZ --> ~P ZZ -> ran ZZ>= C_ ~P ZZ )
3	1, 2	ax-mp 5	. 2 |- ran ZZ>= C_ ~P ZZ
4		ffn 6045	. . . . . 6 |- ( ZZ>= : ZZ --> ~P ZZ -> ZZ>= Fn ZZ )
5	1, 4	ax-mp 5	. . . . 5 |- ZZ>= Fn ZZ
6		1z 11407	. . . . 5 |- 1 e. ZZ
7		fnfvelrn 6356	. . . . 5 |- ( ( ZZ>= Fn ZZ /\ 1 e. ZZ ) -> ( ZZ>= ` 1 ) e. ran ZZ>= )
8	5, 6, 7	mp2an 708	. . . 4 |- ( ZZ>= ` 1 ) e. ran ZZ>=
9	8	ne0ii 3923	. . 3 |- ran ZZ>= =/= (/)
10		uzid 11702	. . . . . . 7 |- ( x e. ZZ -> x e. ( ZZ>= ` x ) )
11		n0i 3920	. . . . . . 7 |- ( x e. ( ZZ>= ` x ) -> -. ( ZZ>= ` x ) = (/) )
12	10, 11	syl 17	. . . . . 6 |- ( x e. ZZ -> -. ( ZZ>= ` x ) = (/) )
13	12	nrex 3000	. . . . 5 |- -. E. x e. ZZ ( ZZ>= ` x ) = (/)
14		fvelrnb 6243	. . . . . 6 |- ( ZZ>= Fn ZZ -> ( (/) e. ran ZZ>= <-> E. x e. ZZ ( ZZ>= ` x ) = (/) ) )
15	5, 14	ax-mp 5	. . . . 5 |- ( (/) e. ran ZZ>= <-> E. x e. ZZ ( ZZ>= ` x ) = (/) )
16	13, 15	mtbir 313	. . . 4 |- -. (/) e. ran ZZ>=
17	16	nelir 2900	. . 3 |- (/) e/ ran ZZ>=
18		uzin2 14084	. . . . 5 |- ( ( x e. ran ZZ>= /\ y e. ran ZZ>= ) -> ( x i^i y ) e. ran ZZ>= )
19		vex 3203	. . . . . . 7 |- x e. _V
20	19	inex1 4799	. . . . . 6 |- ( x i^i y ) e. _V
21	20	pwid 4174	. . . . 5 |- ( x i^i y ) e. ~P ( x i^i y )
22		inelcm 4032	. . . . 5 |- ( ( ( x i^i y ) e. ran ZZ>= /\ ( x i^i y ) e. ~P ( x i^i y ) ) -> ( ran ZZ>= i^i ~P ( x i^i y ) ) =/= (/) )
23	18, 21, 22	sylancl 694	. . . 4 |- ( ( x e. ran ZZ>= /\ y e. ran ZZ>= ) -> ( ran ZZ>= i^i ~P ( x i^i y ) ) =/= (/) )
24	23	rgen2a 2977	. . 3 |- A. x e. ran ZZ>= A. y e. ran ZZ>= ( ran ZZ>= i^i ~P ( x i^i y ) ) =/= (/)
25	9, 17, 24	3pm3.2i 1239	. 2 |- ( ran ZZ>= =/= (/) /\ (/) e/ ran ZZ>= /\ A. x e. ran ZZ>= A. y e. ran ZZ>= ( ran ZZ>= i^i ~P ( x i^i y ) ) =/= (/) )
26		zex 11386	. . 3 |- ZZ e. _V
27		isfbas 21633	. . 3 |- ( ZZ e. _V -> ( ran ZZ>= e. ( fBas ` ZZ ) <-> ( ran ZZ>= C_ ~P ZZ /\ ( ran ZZ>= =/= (/) /\ (/) e/ ran ZZ>= /\ A. x e. ran ZZ>= A. y e. ran ZZ>= ( ran ZZ>= i^i ~P ( x i^i y ) ) =/= (/) ) ) ) )
28	26, 27	ax-mp 5	. 2 |- ( ran ZZ>= e. ( fBas ` ZZ ) <-> ( ran ZZ>= C_ ~P ZZ /\ ( ran ZZ>= =/= (/) /\ (/) e/ ran ZZ>= /\ A. x e. ran ZZ>= A. y e. ran ZZ>= ( ran ZZ>= i^i ~P ( x i^i y ) ) =/= (/) ) ) )
29	3, 25, 28	mpbir2an 955	1 |- ran ZZ>= e. ( fBas ` ZZ )

Syntax hints:   -. wn 3   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   e/ wnel 2897   A.wral 2912   E.wrex 2913   _Vcvv 3200   i^i cin 3573   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   ran crn 5115   Fn wfn 5883   -->wf 5884   ` cfv 5888   1c1 9937   ZZcz 11377   ZZ>=cuz 11687   fBascfbas 19734

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  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  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-reu 2919  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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  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-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-z 11378  df-uz 11688  df-fbas 19743

This theorem is referenced by:  uzfbas  21702