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Theorem zfbas 19444
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.
StepHypRef Expression
1 uzf 10856 . . 3  |-  ZZ>= : ZZ --> ~P ZZ
2 frn 5560 . . 3  |-  ( ZZ>= : ZZ --> ~P ZZ  ->  ran  ZZ>=  C_  ~P ZZ )
31, 2ax-mp 5 . 2  |-  ran  ZZ>=  C_  ~P ZZ
4 ffn 5554 . . . . . 6  |-  ( ZZ>= : ZZ --> ~P ZZ  ->  ZZ>=  Fn  ZZ )
51, 4ax-mp 5 . . . . 5  |-  ZZ>=  Fn  ZZ
6 1z 10668 . . . . 5  |-  1  e.  ZZ
7 fnfvelrn 5835 . . . . 5  |-  ( (
ZZ>=  Fn  ZZ  /\  1  e.  ZZ )  ->  ( ZZ>=
`  1 )  e. 
ran  ZZ>= )
85, 6, 7mp2an 672 . . . 4  |-  ( ZZ>= ` 
1 )  e.  ran  ZZ>=
9 ne0i 3638 . . . 4  |-  ( (
ZZ>= `  1 )  e. 
ran  ZZ>=  ->  ran  ZZ>=  =/=  (/) )
108, 9ax-mp 5 . . 3  |-  ran  ZZ>=  =/=  (/)
11 uzid 10867 . . . . . . 7  |-  ( x  e.  ZZ  ->  x  e.  ( ZZ>= `  x )
)
12 n0i 3637 . . . . . . 7  |-  ( x  e.  ( ZZ>= `  x
)  ->  -.  ( ZZ>=
`  x )  =  (/) )
1311, 12syl 16 . . . . . 6  |-  ( x  e.  ZZ  ->  -.  ( ZZ>= `  x )  =  (/) )
1413nrex 2813 . . . . 5  |-  -.  E. x  e.  ZZ  ( ZZ>=
`  x )  =  (/)
15 fvelrnb 5734 . . . . . 6  |-  ( ZZ>=  Fn  ZZ  ->  ( (/)  e.  ran  ZZ>=  <->  E. x  e.  ZZ  ( ZZ>=
`  x )  =  (/) ) )
165, 15ax-mp 5 . . . . 5  |-  ( (/)  e.  ran  ZZ>= 
<->  E. x  e.  ZZ  ( ZZ>= `  x )  =  (/) )
1714, 16mtbir 299 . . . 4  |-  -.  (/)  e.  ran  ZZ>=
1817nelir 2703 . . 3  |-  (/)  e/  ran  ZZ>=
19 uzin2 12824 . . . . 5  |-  ( ( x  e.  ran  ZZ>=  /\  y  e.  ran  ZZ>= )  -> 
( x  i^i  y
)  e.  ran  ZZ>= )
20 vex 2970 . . . . . . 7  |-  x  e. 
_V
2120inex1 4428 . . . . . 6  |-  ( x  i^i  y )  e. 
_V
2221pwid 3869 . . . . 5  |-  ( x  i^i  y )  e. 
~P ( x  i^i  y )
23 inelcm 3728 . . . . 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 ) )  =/=  (/) )
2419, 22, 23sylancl 662 . . . 4  |-  ( ( x  e.  ran  ZZ>=  /\  y  e.  ran  ZZ>= )  -> 
( ran  ZZ>=  i^i  ~P ( x  i^i  y
) )  =/=  (/) )
2524rgen2a 2777 . . 3  |-  A. x  e.  ran  ZZ>= A. y  e.  ran  ZZ>= ( ran  ZZ>=  i^i  ~P (
x  i^i  y )
)  =/=  (/)
2610, 18, 253pm3.2i 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 10647 . . 3  |-  ZZ  e.  _V
28 isfbas 19377 . . 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 )
)  =/=  (/) ) ) ) )
2927, 28ax-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 ) )  =/=  (/) ) ) )
303, 26, 29mpbir2an 911 1  |-  ran  ZZ>=  e.  ( fBas `  ZZ )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2601    e/ wnel 2602   A.wral 2710   E.wrex 2711   _Vcvv 2967    i^i cin 3322    C_ wss 3323   (/)c0 3632   ~Pcpw 3855   ran crn 4836    Fn wfn 5408   -->wf 5409   ` cfv 5413   1c1 9275   ZZcz 10638   ZZ>=cuz 10853   fBascfbas 17779
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-recs 6824  df-rdg 6858  df-er 7093  df-en 7303  df-dom 7304  df-sdom 7305  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-nn 10315  df-z 10639  df-uz 10854  df-fbas 17789
This theorem is referenced by:  uzfbas  19446
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