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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.
StepHypRef Expression
1 uzf 11191 . . 3  |-  ZZ>= : ZZ --> ~P ZZ
2 frn 5758 . . 3  |-  ( ZZ>= : ZZ --> ~P ZZ  ->  ran  ZZ>=  C_  ~P ZZ )
31, 2ax-mp 5 . 2  |-  ran  ZZ>=  C_  ~P ZZ
4 ffn 5751 . . . . . 6  |-  ( ZZ>= : ZZ --> ~P ZZ  ->  ZZ>=  Fn  ZZ )
51, 4ax-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>= )
85, 6, 7mp2an 683 . . . 4  |-  ( ZZ>= ` 
1 )  e.  ran  ZZ>=
98ne0ii 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 )  =  (/) )
1210, 11syl 17 . . . . . 6  |-  ( x  e.  ZZ  ->  -.  ( ZZ>= `  x )  =  (/) )
1312nrex 2854 . . . . 5  |-  -.  E. x  e.  ZZ  ( ZZ>=
`  x )  =  (/)
14 fvelrnb 5935 . . . . . 6  |-  ( ZZ>=  Fn  ZZ  ->  ( (/)  e.  ran  ZZ>=  <->  E. x  e.  ZZ  ( ZZ>=
`  x )  =  (/) ) )
155, 14ax-mp 5 . . . . 5  |-  ( (/)  e.  ran  ZZ>= 
<->  E. x  e.  ZZ  ( ZZ>= `  x )  =  (/) )
1613, 15mtbir 305 . . . 4  |-  -.  (/)  e.  ran  ZZ>=
1716nelir 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
2019inex1 4558 . . . . . 6  |-  ( x  i^i  y )  e. 
_V
2120pwid 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 ) )  =/=  (/) )
2318, 21, 22sylancl 673 . . . 4  |-  ( ( x  e.  ran  ZZ>=  /\  y  e.  ran  ZZ>= )  -> 
( ran  ZZ>=  i^i  ~P ( x  i^i  y
) )  =/=  (/) )
2423rgen2a 2827 . . 3  |-  A. x  e.  ran  ZZ>= A. y  e.  ran  ZZ>= ( ran  ZZ>=  i^i  ~P (
x  i^i  y )
)  =/=  (/)
259, 17, 243pm3.2i 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 )
)  =/=  (/) ) ) ) )
2826, 27ax-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 ) )  =/=  (/) ) ) )
293, 25, 28mpbir2an 936 1  |-  ran  ZZ>=  e.  ( fBas `  ZZ )
Colors of variables: wff setvar class
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
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