Theorem zfbas 19600

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 10974	. . 3 |- ZZ>= : ZZ --> ~P ZZ
2		frn 5672	. . 3 |- ( ZZ>= : ZZ --> ~P ZZ -> ran ZZ>= C_ ~P ZZ )
3	1, 2	ax-mp 5	. 2 |- ran ZZ>= C_ ~P ZZ
4		ffn 5666	. . . . . 6 |- ( ZZ>= : ZZ --> ~P ZZ -> ZZ>= Fn ZZ )
5	1, 4	ax-mp 5	. . . . 5 |- ZZ>= Fn ZZ
6		1z 10786	. . . . 5 |- 1 e. ZZ
7		fnfvelrn 5948	. . . . 5 |- ( ( ZZ>= Fn ZZ /\ 1 e. ZZ ) -> ( ZZ>= ` 1 ) e. ran ZZ>= )
8	5, 6, 7	mp2an 672	. . . 4 |- ( ZZ>= ` 1 ) e. ran ZZ>=
9		ne0i 3750	. . . 4 |- ( ( ZZ>= ` 1 ) e. ran ZZ>= -> ran ZZ>= =/= (/) )
10	8, 9	ax-mp 5	. . 3 |- ran ZZ>= =/= (/)
11		uzid 10985	. . . . . . 7 |- ( x e. ZZ -> x e. ( ZZ>= ` x ) )
12		n0i 3749	. . . . . . 7 |- ( x e. ( ZZ>= ` x ) -> -. ( ZZ>= ` x ) = (/) )
13	11, 12	syl 16	. . . . . 6 |- ( x e. ZZ -> -. ( ZZ>= ` x ) = (/) )
14	13	nrex 2922	. . . . 5 |- -. E. x e. ZZ ( ZZ>= ` x ) = (/)
15		fvelrnb 5847	. . . . . 6 |- ( ZZ>= Fn ZZ -> ( (/) e. ran ZZ>= <-> E. x e. ZZ ( ZZ>= ` x ) = (/) ) )
16	5, 15	ax-mp 5	. . . . 5 |- ( (/) e. ran ZZ>= <-> E. x e. ZZ ( ZZ>= ` x ) = (/) )
17	14, 16	mtbir 299	. . . 4 |- -. (/) e. ran ZZ>=
18	17	nelir 2787	. . 3 |- (/) e/ ran ZZ>=
19		uzin2 12949	. . . . 5 |- ( ( x e. ran ZZ>= /\ y e. ran ZZ>= ) -> ( x i^i y ) e. ran ZZ>= )
20		vex 3079	. . . . . . 7 |- x e. _V
21	20	inex1 4540	. . . . . 6 |- ( x i^i y ) e. _V
22	21	pwid 3981	. . . . 5 |- ( x i^i y ) e. ~P ( x i^i y )
23		inelcm 3840	. . . . 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 ) ) =/= (/) )
24	19, 22, 23	sylancl 662	. . . 4 |- ( ( x e. ran ZZ>= /\ y e. ran ZZ>= ) -> ( ran ZZ>= i^i ~P ( x i^i y ) ) =/= (/) )
25	24	rgen2a 2898	. . 3 |- A. x e. ran ZZ>= A. y e. ran ZZ>= ( ran ZZ>= i^i ~P ( x i^i y ) ) =/= (/)
26	10, 18, 25	3pm3.2i 1166	. 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 ) ) =/= (/) )
27		zex 10765	. . 3 |- ZZ e. _V
28		isfbas 19533	. . 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 ) ) =/= (/) ) ) ) )
29	27, 28	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 ) ) =/= (/) ) ) )
30	3, 26, 29	mpbir2an 911	1 |- ran ZZ>= e. ( fBas ` ZZ )

Syntax hints:   -. wn 3   <-> wb 184   /\ wa 369   /\ w3a 965   = wceq 1370   e. wcel 1758   =/= wne 2647   e/ wnel 2648   A.wral 2798   E.wrex 2799   _Vcvv 3076   i^i cin 3434   C_ wss 3435   (/)c0 3744   ~Pcpw 3967   ran crn 4948   Fn wfn 5520   -->wf 5521   ` cfv 5525   1c1 9393   ZZcz 10756   ZZ>=cuz 10971   fBascfbas 17928

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638  ax-un 6481  ax-cnex 9448  ax-resscn 9449  ax-1cn 9450  ax-icn 9451  ax-addcl 9452  ax-addrcl 9453  ax-mulcl 9454  ax-mulrcl 9455  ax-mulcom 9456  ax-addass 9457  ax-mulass 9458  ax-distr 9459  ax-i2m1 9460  ax-1ne0 9461  ax-1rid 9462  ax-rnegex 9463  ax-rrecex 9464  ax-cnre 9465  ax-pre-lttri 9466  ax-pre-lttrn 9467  ax-pre-ltadd 9468  ax-pre-mulgt0 9469

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-nel 2650  df-ral 2803  df-rex 2804  df-reu 2805  df-rab 2807  df-v 3078  df-sbc 3293  df-csb 3395  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-pss 3451  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-tp 3989  df-op 3991  df-uni 4199  df-iun 4280  df-br 4400  df-opab 4458  df-mpt 4459  df-tr 4493  df-eprel 4739  df-id 4743  df-po 4748  df-so 4749  df-fr 4786  df-we 4788  df-ord 4829  df-on 4830  df-lim 4831  df-suc 4832  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-riota 6160  df-ov 6202  df-oprab 6203  df-mpt2 6204  df-om 6586  df-recs 6941  df-rdg 6975  df-er 7210  df-en 7420  df-dom 7421  df-sdom 7422  df-pnf 9530  df-mnf 9531  df-xr 9532  df-ltxr 9533  df-le 9534  df-sub 9707  df-neg 9708  df-nn 10433  df-z 10757  df-uz 10972  df-fbas 17938

This theorem is referenced by:  uzfbas  19602