Theorem zfbas 20523

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 11109	. . 3 |- ZZ>= : ZZ --> ~P ZZ
2		frn 5743	. . 3 |- ( ZZ>= : ZZ --> ~P ZZ -> ran ZZ>= C_ ~P ZZ )
3	1, 2	ax-mp 5	. 2 |- ran ZZ>= C_ ~P ZZ
4		ffn 5737	. . . . . 6 |- ( ZZ>= : ZZ --> ~P ZZ -> ZZ>= Fn ZZ )
5	1, 4	ax-mp 5	. . . . 5 |- ZZ>= Fn ZZ
6		1z 10915	. . . . 5 |- 1 e. ZZ
7		fnfvelrn 6029	. . . . 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	8	ne0ii 3800	. . 3 |- ran ZZ>= =/= (/)
10		uzid 11120	. . . . . . 7 |- ( x e. ZZ -> x e. ( ZZ>= ` x ) )
11		n0i 3798	. . . . . . 7 |- ( x e. ( ZZ>= ` x ) -> -. ( ZZ>= ` x ) = (/) )
12	10, 11	syl 16	. . . . . 6 |- ( x e. ZZ -> -. ( ZZ>= ` x ) = (/) )
13	12	nrex 2912	. . . . 5 |- -. E. x e. ZZ ( ZZ>= ` x ) = (/)
14		fvelrnb 5920	. . . . . 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 299	. . . 4 |- -. (/) e. ran ZZ>=
17	16	nelir 2793	. . 3 |- (/) e/ ran ZZ>=
18		uzin2 13189	. . . . 5 |- ( ( x e. ran ZZ>= /\ y e. ran ZZ>= ) -> ( x i^i y ) e. ran ZZ>= )
19		vex 3112	. . . . . . 7 |- x e. _V
20	19	inex1 4597	. . . . . 6 |- ( x i^i y ) e. _V
21	20	pwid 4029	. . . . 5 |- ( x i^i y ) e. ~P ( x i^i y )
22		inelcm 3884	. . . . 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 662	. . . 4 |- ( ( x e. ran ZZ>= /\ y e. ran ZZ>= ) -> ( ran ZZ>= i^i ~P ( x i^i y ) ) =/= (/) )
24	23	rgen2a 2884	. . 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 1174	. 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 10894	. . 3 |- ZZ e. _V
27		isfbas 20456	. . 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 920	1 |- ran ZZ>= e. ( fBas ` ZZ )

Syntax hints:   -. wn 3   <-> wb 184   /\ wa 369   /\ w3a 973   = wceq 1395   e. wcel 1819   =/= wne 2652   e/ wnel 2653   A.wral 2807   E.wrex 2808   _Vcvv 3109   i^i cin 3470   C_ wss 3471   (/)c0 3793   ~Pcpw 4015   ran crn 5009   Fn wfn 5589   -->wf 5590   ` cfv 5594   1c1 9510   ZZcz 10885   ZZ>=cuz 11106   fBascfbas 18533

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-recs 7060  df-rdg 7094  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-z 10886  df-uz 11107  df-fbas 18543

This theorem is referenced by:  uzfbas  20525