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Theorem zfbas 20523
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 11109 . . 3  |-  ZZ>= : ZZ --> ~P ZZ
2 frn 5743 . . 3  |-  ( ZZ>= : ZZ --> ~P ZZ  ->  ran  ZZ>=  C_  ~P ZZ )
31, 2ax-mp 5 . 2  |-  ran  ZZ>=  C_  ~P ZZ
4 ffn 5737 . . . . . 6  |-  ( ZZ>= : ZZ --> ~P ZZ  ->  ZZ>=  Fn  ZZ )
51, 4ax-mp 5 . . . . 5  |-  ZZ>=  Fn  ZZ
6 1z 10915 . . . . 5  |-  1  e.  ZZ
7 fnfvelrn 6029 . . . . 5  |-  ( (
ZZ>=  Fn  ZZ  /\  1  e.  ZZ )  ->  ( ZZ>=
`  1 )  e. 
ran  ZZ>= )
85, 6, 7mp2an 672 . . . 4  |-  ( ZZ>= ` 
1 )  e.  ran  ZZ>=
98ne0ii 3800 . . 3  |-  ran  ZZ>=  =/=  (/)
10 uzid 11120 . . . . . . 7  |-  ( x  e.  ZZ  ->  x  e.  ( ZZ>= `  x )
)
11 n0i 3798 . . . . . . 7  |-  ( x  e.  ( ZZ>= `  x
)  ->  -.  ( ZZ>=
`  x )  =  (/) )
1210, 11syl 16 . . . . . 6  |-  ( x  e.  ZZ  ->  -.  ( ZZ>= `  x )  =  (/) )
1312nrex 2912 . . . . 5  |-  -.  E. x  e.  ZZ  ( ZZ>=
`  x )  =  (/)
14 fvelrnb 5920 . . . . . 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 299 . . . 4  |-  -.  (/)  e.  ran  ZZ>=
1716nelir 2793 . . 3  |-  (/)  e/  ran  ZZ>=
18 uzin2 13189 . . . . 5  |-  ( ( x  e.  ran  ZZ>=  /\  y  e.  ran  ZZ>= )  -> 
( x  i^i  y
)  e.  ran  ZZ>= )
19 vex 3112 . . . . . . 7  |-  x  e. 
_V
2019inex1 4597 . . . . . 6  |-  ( x  i^i  y )  e. 
_V
2120pwid 4029 . . . . 5  |-  ( x  i^i  y )  e. 
~P ( x  i^i  y )
22 inelcm 3884 . . . . 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 662 . . . 4  |-  ( ( x  e.  ran  ZZ>=  /\  y  e.  ran  ZZ>= )  -> 
( ran  ZZ>=  i^i  ~P ( x  i^i  y
) )  =/=  (/) )
2423rgen2a 2884 . . 3  |-  A. x  e.  ran  ZZ>= A. y  e.  ran  ZZ>= ( ran  ZZ>=  i^i  ~P (
x  i^i  y )
)  =/=  (/)
259, 17, 243pm3.2i 1174 . 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 10894 . . 3  |-  ZZ  e.  _V
27 isfbas 20456 . . 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 920 1  |-  ran  ZZ>=  e.  ( fBas `  ZZ )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819    =/= wne 2652    e/ wnel 2653   A.wral 2807   E.wrex 2808   _Vcvv 3109    i^i cin 3470    C_ wss 3471   (/)c0 3793   ~Pcpw 4015   ran crn 5009    Fn wfn 5589   -->wf 5590   ` cfv 5594   1c1 9510   ZZcz 10885   ZZ>=cuz 11106   fBascfbas 18533
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-recs 7060  df-rdg 7094  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-z 10886  df-uz 11107  df-fbas 18543
This theorem is referenced by:  uzfbas  20525
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