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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.
StepHypRef Expression
1 uzf 10974 . . 3  |-  ZZ>= : ZZ --> ~P ZZ
2 frn 5672 . . 3  |-  ( ZZ>= : ZZ --> ~P ZZ  ->  ran  ZZ>=  C_  ~P ZZ )
31, 2ax-mp 5 . 2  |-  ran  ZZ>=  C_  ~P ZZ
4 ffn 5666 . . . . . 6  |-  ( ZZ>= : ZZ --> ~P ZZ  ->  ZZ>=  Fn  ZZ )
51, 4ax-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>= )
85, 6, 7mp2an 672 . . . 4  |-  ( ZZ>= ` 
1 )  e.  ran  ZZ>=
9 ne0i 3750 . . . 4  |-  ( (
ZZ>= `  1 )  e. 
ran  ZZ>=  ->  ran  ZZ>=  =/=  (/) )
108, 9ax-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 )  =  (/) )
1311, 12syl 16 . . . . . 6  |-  ( x  e.  ZZ  ->  -.  ( ZZ>= `  x )  =  (/) )
1413nrex 2922 . . . . 5  |-  -.  E. x  e.  ZZ  ( ZZ>=
`  x )  =  (/)
15 fvelrnb 5847 . . . . . 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 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
2120inex1 4540 . . . . . 6  |-  ( x  i^i  y )  e. 
_V
2221pwid 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 ) )  =/=  (/) )
2419, 22, 23sylancl 662 . . . 4  |-  ( ( x  e.  ran  ZZ>=  /\  y  e.  ran  ZZ>= )  -> 
( ran  ZZ>=  i^i  ~P ( x  i^i  y
) )  =/=  (/) )
2524rgen2a 2898 . . 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 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 )
)  =/=  (/) ) ) ) )
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 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
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