Theorem zfbas 20960

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 11191	. . 3 |- ZZ>= : ZZ --> ~P ZZ
2		frn 5758	. . 3 |- ( ZZ>= : ZZ --> ~P ZZ -> ran ZZ>= C_ ~P ZZ )
3	1, 2	ax-mp 5	. 2 |- ran ZZ>= C_ ~P ZZ
4		ffn 5751	. . . . . 6 |- ( ZZ>= : ZZ --> ~P ZZ -> ZZ>= Fn ZZ )
5	1, 4	ax-mp 5	. . . . 5 |- ZZ>= Fn ZZ
6		1z 10996	. . . . 5 |- 1 e. ZZ
7		fnfvelrn 6042	. . . . 5 |- ( ( ZZ>= Fn ZZ /\ 1 e. ZZ ) -> ( ZZ>= ` 1 ) e. ran ZZ>= )
8	5, 6, 7	mp2an 683	. . . 4 |- ( ZZ>= ` 1 ) e. ran ZZ>=
9	8	ne0ii 3750	. . 3 |- ran ZZ>= =/= (/)
10		uzid 11202	. . . . . . 7 |- ( x e. ZZ -> x e. ( ZZ>= ` x ) )
11		n0i 3748	. . . . . . 7 |- ( x e. ( ZZ>= ` x ) -> -. ( ZZ>= ` x ) = (/) )
12	10, 11	syl 17	. . . . . 6 |- ( x e. ZZ -> -. ( ZZ>= ` x ) = (/) )
13	12	nrex 2854	. . . . 5 |- -. E. x e. ZZ ( ZZ>= ` x ) = (/)
14		fvelrnb 5935	. . . . . 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 305	. . . 4 |- -. (/) e. ran ZZ>=
17	16	nelir 2739	. . 3 |- (/) e/ ran ZZ>=
18		uzin2 13456	. . . . 5 |- ( ( x e. ran ZZ>= /\ y e. ran ZZ>= ) -> ( x i^i y ) e. ran ZZ>= )
19		vex 3060	. . . . . . 7 |- x e. _V
20	19	inex1 4558	. . . . . 6 |- ( x i^i y ) e. _V
21	20	pwid 3977	. . . . 5 |- ( x i^i y ) e. ~P ( x i^i y )
22		inelcm 3831	. . . . 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 673	. . . 4 |- ( ( x e. ran ZZ>= /\ y e. ran ZZ>= ) -> ( ran ZZ>= i^i ~P ( x i^i y ) ) =/= (/) )
24	23	rgen2a 2827	. . 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 1192	. 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 10975	. . 3 |- ZZ e. _V
27		isfbas 20893	. . 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 936	1 |- ran ZZ>= e. ( fBas ` ZZ )

Syntax hints:   -. wn 3   <-> wb 189   /\ wa 375   /\ w3a 991   = wceq 1455   e. wcel 1898   =/= wne 2633   e/ wnel 2634   A.wral 2749   E.wrex 2750   _Vcvv 3057   i^i cin 3415   C_ wss 3416   (/)c0 3743   ~Pcpw 3963   ran crn 4854   Fn wfn 5596   -->wf 5597   ` cfv 5601   1c1 9566   ZZcz 10966   ZZ>=cuz 11188   fBascfbas 19007

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4539  ax-nul 4548  ax-pow 4595  ax-pr 4653  ax-un 6610  ax-cnex 9621  ax-resscn 9622  ax-1cn 9623  ax-icn 9624  ax-addcl 9625  ax-addrcl 9626  ax-mulcl 9627  ax-mulrcl 9628  ax-mulcom 9629  ax-addass 9630  ax-mulass 9631  ax-distr 9632  ax-i2m1 9633  ax-1ne0 9634  ax-1rid 9635  ax-rnegex 9636  ax-rrecex 9637  ax-cnre 9638  ax-pre-lttri 9639  ax-pre-lttrn 9640  ax-pre-ltadd 9641  ax-pre-mulgt0 9642

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-nel 2636  df-ral 2754  df-rex 2755  df-reu 2756  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4213  df-iun 4294  df-br 4417  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6277  df-ov 6318  df-oprab 6319  df-mpt2 6320  df-om 6720  df-wrecs 7054  df-recs 7116  df-rdg 7154  df-er 7389  df-en 7596  df-dom 7597  df-sdom 7598  df-pnf 9703  df-mnf 9704  df-xr 9705  df-ltxr 9706  df-le 9707  df-sub 9888  df-neg 9889  df-nn 10638  df-z 10967  df-uz 11189  df-fbas 19016

This theorem is referenced by:  uzfbas  20962