Theorem zfbas 17881

Description: The set of upper integer sets 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 10447	. . 3 |- ZZ>= : ZZ --> ~P ZZ
2		frn 5556	. . 3 |- ( ZZ>= : ZZ --> ~P ZZ -> ran ZZ>= C_ ~P ZZ )
3	1, 2	ax-mp 8	. 2 |- ran ZZ>= C_ ~P ZZ
4		ffn 5550	. . . . . 6 |- ( ZZ>= : ZZ --> ~P ZZ -> ZZ>= Fn ZZ )
5	1, 4	ax-mp 8	. . . . 5 |- ZZ>= Fn ZZ
6		1z 10267	. . . . 5 |- 1 e. ZZ
7		fnfvelrn 5826	. . . . 5 |- ( ( ZZ>= Fn ZZ /\ 1 e. ZZ ) -> ( ZZ>= ` 1 ) e. ran ZZ>= )
8	5, 6, 7	mp2an 654	. . . 4 |- ( ZZ>= ` 1 ) e. ran ZZ>=
9		ne0i 3594	. . . 4 |- ( ( ZZ>= ` 1 ) e. ran ZZ>= -> ran ZZ>= =/= (/) )
10	8, 9	ax-mp 8	. . 3 |- ran ZZ>= =/= (/)
11		uzid 10456	. . . . . . 7 |- ( x e. ZZ -> x e. ( ZZ>= ` x ) )
12		n0i 3593	. . . . . . 7 |- ( x e. ( ZZ>= ` x ) -> -. ( ZZ>= ` x ) = (/) )
13	11, 12	syl 16	. . . . . 6 |- ( x e. ZZ -> -. ( ZZ>= ` x ) = (/) )
14	13	nrex 2768	. . . . 5 |- -. E. x e. ZZ ( ZZ>= ` x ) = (/)
15		fvelrnb 5733	. . . . . 6 |- ( ZZ>= Fn ZZ -> ( (/) e. ran ZZ>= <-> E. x e. ZZ ( ZZ>= ` x ) = (/) ) )
16	5, 15	ax-mp 8	. . . . 5 |- ( (/) e. ran ZZ>= <-> E. x e. ZZ ( ZZ>= ` x ) = (/) )
17	14, 16	mtbir 291	. . . 4 |- -. (/) e. ran ZZ>=
18		df-nel 2570	. . . 4 |- ( (/) e/ ran ZZ>= <-> -. (/) e. ran ZZ>= )
19	17, 18	mpbir 201	. . 3 |- (/) e/ ran ZZ>=
20		uzin2 12103	. . . . 5 |- ( ( x e. ran ZZ>= /\ y e. ran ZZ>= ) -> ( x i^i y ) e. ran ZZ>= )
21		vex 2919	. . . . . . 7 |- x e. _V
22	21	inex1 4304	. . . . . 6 |- ( x i^i y ) e. _V
23	22	pwid 3772	. . . . 5 |- ( x i^i y ) e. ~P ( x i^i y )
24		inelcm 3642	. . . . 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 ) ) =/= (/) )
25	20, 23, 24	sylancl 644	. . . 4 |- ( ( x e. ran ZZ>= /\ y e. ran ZZ>= ) -> ( ran ZZ>= i^i ~P ( x i^i y ) ) =/= (/) )
26	25	rgen2a 2732	. . 3 |- A. x e. ran ZZ>= A. y e. ran ZZ>= ( ran ZZ>= i^i ~P ( x i^i y ) ) =/= (/)
27	10, 19, 26	3pm3.2i 1132	. 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 ) ) =/= (/) )
28		zex 10247	. . 3 |- ZZ e. _V
29		isfbas 17814	. . 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 ) ) =/= (/) ) ) ) )
30	28, 29	ax-mp 8	. 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 ) ) =/= (/) ) ) )
31	3, 27, 30	mpbir2an 887	1 |- ran ZZ>= e. ( fBas ` ZZ )

Syntax hints:   -. wn 3   <-> wb 177   /\ wa 359   /\ w3a 936   = wceq 1649   e. wcel 1721   =/= wne 2567   e/ wnel 2568   A.wral 2666   E.wrex 2667   _Vcvv 2916   i^i cin 3279   C_ wss 3280   (/)c0 3588   ~Pcpw 3759   ran crn 4838   Fn wfn 5408   -->wf 5409   ` cfv 5413   1c1 8947   ZZcz 10238   ZZ>=cuz 10444   fBascfbas 16644

This theorem is referenced by:  uzfbas  17883

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-riota 6508  df-recs 6592  df-rdg 6627  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-z 10239  df-uz 10445  df-fbas 16654