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Theorem zfbas 17881
Description: The set of upper integer sets 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 10447 . . 3  |-  ZZ>= : ZZ --> ~P ZZ
2 frn 5556 . . 3  |-  ( ZZ>= : ZZ --> ~P ZZ  ->  ran  ZZ>=  C_  ~P ZZ )
31, 2ax-mp 8 . 2  |-  ran  ZZ>=  C_  ~P ZZ
4 ffn 5550 . . . . . 6  |-  ( ZZ>= : ZZ --> ~P ZZ  ->  ZZ>=  Fn  ZZ )
51, 4ax-mp 8 . . . . 5  |-  ZZ>=  Fn  ZZ
6 1z 10267 . . . . 5  |-  1  e.  ZZ
7 fnfvelrn 5826 . . . . 5  |-  ( (
ZZ>=  Fn  ZZ  /\  1  e.  ZZ )  ->  ( ZZ>=
`  1 )  e. 
ran  ZZ>= )
85, 6, 7mp2an 654 . . . 4  |-  ( ZZ>= ` 
1 )  e.  ran  ZZ>=
9 ne0i 3594 . . . 4  |-  ( (
ZZ>= `  1 )  e. 
ran  ZZ>=  ->  ran  ZZ>=  =/=  (/) )
108, 9ax-mp 8 . . 3  |-  ran  ZZ>=  =/=  (/)
11 uzid 10456 . . . . . . 7  |-  ( x  e.  ZZ  ->  x  e.  ( ZZ>= `  x )
)
12 n0i 3593 . . . . . . 7  |-  ( x  e.  ( ZZ>= `  x
)  ->  -.  ( ZZ>=
`  x )  =  (/) )
1311, 12syl 16 . . . . . 6  |-  ( x  e.  ZZ  ->  -.  ( ZZ>= `  x )  =  (/) )
1413nrex 2768 . . . . 5  |-  -.  E. x  e.  ZZ  ( ZZ>=
`  x )  =  (/)
15 fvelrnb 5733 . . . . . 6  |-  ( ZZ>=  Fn  ZZ  ->  ( (/)  e.  ran  ZZ>=  <->  E. x  e.  ZZ  ( ZZ>=
`  x )  =  (/) ) )
165, 15ax-mp 8 . . . . 5  |-  ( (/)  e.  ran  ZZ>= 
<->  E. x  e.  ZZ  ( ZZ>= `  x )  =  (/) )
1714, 16mtbir 291 . . . 4  |-  -.  (/)  e.  ran  ZZ>=
18 df-nel 2570 . . . 4  |-  ( (/)  e/ 
ran  ZZ>= 
<->  -.  (/)  e.  ran  ZZ>= )
1917, 18mpbir 201 . . 3  |-  (/)  e/  ran  ZZ>=
20 uzin2 12103 . . . . 5  |-  ( ( x  e.  ran  ZZ>=  /\  y  e.  ran  ZZ>= )  -> 
( x  i^i  y
)  e.  ran  ZZ>= )
21 vex 2919 . . . . . . 7  |-  x  e. 
_V
2221inex1 4304 . . . . . 6  |-  ( x  i^i  y )  e. 
_V
2322pwid 3772 . . . . 5  |-  ( x  i^i  y )  e. 
~P ( x  i^i  y )
24 inelcm 3642 . . . . 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 ) )  =/=  (/) )
2520, 23, 24sylancl 644 . . . 4  |-  ( ( x  e.  ran  ZZ>=  /\  y  e.  ran  ZZ>= )  -> 
( ran  ZZ>=  i^i  ~P ( x  i^i  y
) )  =/=  (/) )
2625rgen2a 2732 . . 3  |-  A. x  e.  ran  ZZ>= A. y  e.  ran  ZZ>= ( ran  ZZ>=  i^i  ~P (
x  i^i  y )
)  =/=  (/)
2710, 19, 263pm3.2i 1132 . 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 )
)  =/=  (/) )
28 zex 10247 . . 3  |-  ZZ  e.  _V
29 isfbas 17814 . . 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 )
)  =/=  (/) ) ) ) )
3028, 29ax-mp 8 . 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 ) )  =/=  (/) ) ) )
313, 27, 30mpbir2an 887 1  |-  ran  ZZ>=  e.  ( fBas `  ZZ )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2567    e/ wnel 2568   A.wral 2666   E.wrex 2667   _Vcvv 2916    i^i cin 3279    C_ wss 3280   (/)c0 3588   ~Pcpw 3759   ran crn 4838    Fn wfn 5408   -->wf 5409   ` cfv 5413   1c1 8947   ZZcz 10238   ZZ>=cuz 10444   fBascfbas 16644
This theorem is referenced by:  uzfbas  17883
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-riota 6508  df-recs 6592  df-rdg 6627  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-z 10239  df-uz 10445  df-fbas 16654
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