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Theorem fsubbas 21671
Description: A condition for a set to generate a filter base. (Contributed by Jeff Hankins, 2-Sep-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)
Assertion
Ref Expression
fsubbas |- ( X e. V -> ( ( fi ` A ) e. ( fBas ` X ) <-> ( A C_ ~P X /\ A =/= (/) /\ -. (/) e. ( fi ` A ) ) ) )
Proof of Theorem fsubbas
Dummy variables x y z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fbasne0 21634 . . . . . 6 |- ( ( fi ` A ) e. ( fBas ` X ) -> ( fi ` A ) =/= (/) )
2 fvprc 6185 . . . . . . 7 |- ( -. A e. _V -> ( fi ` A ) = (/) )
32necon1ai 2821 . . . . . 6 |- ( ( fi ` A ) =/= (/) -> A e. _V )
41, 3syl 17 . . . . 5 |- ( ( fi ` A ) e. ( fBas ` X ) -> A e. _V )
5 ssfii 8325 . . . . 5 |- ( A e. _V -> A C_ ( fi ` A ) )
64, 5syl 17 . . . 4 |- ( ( fi ` A ) e. ( fBas ` X ) -> A C_ ( fi ` A ) )
7 fbsspw 21636 . . . 4 |- ( ( fi ` A ) e. ( fBas ` X ) -> ( fi ` A ) C_ ~P X )
86, 7sstrd 3613 . . 3 |- ( ( fi ` A ) e. ( fBas ` X ) -> A C_ ~P X )
9 fieq0 8327 . . . . . 6 |- ( A e. _V -> ( A = (/) <-> ( fi ` A ) = (/) ) )
109necon3bid 2838 . . . . 5 |- ( A e. _V -> ( A =/= (/) <-> ( fi ` A ) =/= (/) ) )
1110biimpar 502 . . . 4 |- ( ( A e. _V /\ ( fi ` A ) =/= (/) ) -> A =/= (/) )
124, 1, 11syl2anc 693 . . 3 |- ( ( fi ` A ) e. ( fBas ` X ) -> A =/= (/) )
13 0nelfb 21635 . . 3 |- ( ( fi ` A ) e. ( fBas ` X ) -> -. (/) e. ( fi ` A ) )
148, 12, 133jca 1242 . 2 |- ( ( fi ` A ) e. ( fBas ` X ) -> ( A C_ ~P X /\ A =/= (/) /\ -. (/) e. ( fi ` A ) ) )
15 simpr1 1067 . . . . 5 |- ( ( X e. V /\ ( A C_ ~P X /\ A =/= (/) /\ -. (/) e. ( fi ` A ) ) ) -> A C_ ~P X )
16 fipwss 8335 . . . . 5 |- ( A C_ ~P X -> ( fi ` A ) C_ ~P X )
1715, 16syl 17 . . . 4 |- ( ( X e. V /\ ( A C_ ~P X /\ A =/= (/) /\ -. (/) e. ( fi ` A ) ) ) -> ( fi ` A ) C_ ~P X )
18 pwexg 4850 . . . . . . . 8 |- ( X e. V -> ~P X e. _V )
1918adantr 481 . . . . . . 7 |- ( ( X e. V /\ ( A C_ ~P X /\ A =/= (/) /\ -. (/) e. ( fi ` A ) ) ) -> ~P X e. _V )
2019, 15ssexd 4805 . . . . . 6 |- ( ( X e. V /\ ( A C_ ~P X /\ A =/= (/) /\ -. (/) e. ( fi ` A ) ) ) -> A e. _V )
21 simpr2 1068 . . . . . 6 |- ( ( X e. V /\ ( A C_ ~P X /\ A =/= (/) /\ -. (/) e. ( fi ` A ) ) ) -> A =/= (/) )
2210biimpa 501 . . . . . 6 |- ( ( A e. _V /\ A =/= (/) ) -> ( fi ` A ) =/= (/) )
2320, 21, 22syl2anc 693 . . . . 5 |- ( ( X e. V /\ ( A C_ ~P X /\ A =/= (/) /\ -. (/) e. ( fi ` A ) ) ) -> ( fi ` A ) =/= (/) )
24 simpr3 1069 . . . . . 6 |- ( ( X e. V /\ ( A C_ ~P X /\ A =/= (/) /\ -. (/) e. ( fi ` A ) ) ) -> -. (/) e. ( fi ` A ) )
25 df-nel 2898 . . . . . 6 |- ( (/) e/ ( fi ` A ) <-> -. (/) e. ( fi ` A ) )
2624, 25sylibr 224 . . . . 5 |- ( ( X e. V /\ ( A C_ ~P X /\ A =/= (/) /\ -. (/) e. ( fi ` A ) ) ) -> (/) e/ ( fi ` A ) )
27 fiin 8328 . . . . . . . 8 |- ( ( x e. ( fi ` A ) /\ y e. ( fi ` A ) ) -> ( x i^i y ) e. ( fi ` A ) )
28 ssid 3624 . . . . . . . 8 |- ( x i^i y ) C_ ( x i^i y )
29 sseq1 3626 . . . . . . . . 9 |- ( z = ( x i^i y ) -> ( z C_ ( x i^i y ) <-> ( x i^i y ) C_ ( x i^i y ) ) )
3029rspcev 3309 . . . . . . . 8 |- ( ( ( x i^i y ) e. ( fi ` A ) /\ ( x i^i y ) C_ ( x i^i y ) ) -> E. z e. ( fi ` A ) z C_ ( x i^i y ) )
3127, 28, 30sylancl 694 . . . . . . 7 |- ( ( x e. ( fi ` A ) /\ y e. ( fi ` A ) ) -> E. z e. ( fi ` A ) z C_ ( x i^i y ) )
3231rgen2a 2977 . . . . . 6 |- A. x e. ( fi ` A ) A. y e. ( fi ` A ) E. z e. ( fi ` A ) z C_ ( x i^i y )
3332a1i 11 . . . . 5 |- ( ( X e. V /\ ( A C_ ~P X /\ A =/= (/) /\ -. (/) e. ( fi ` A ) ) ) -> A. x e. ( fi ` A ) A. y e. ( fi ` A ) E. z e. ( fi ` A ) z C_ ( x i^i y ) )
3423, 26, 333jca 1242 . . . 4 |- ( ( X e. V /\ ( A C_ ~P X /\ A =/= (/) /\ -. (/) e. ( fi ` A ) ) ) -> ( ( fi ` A ) =/= (/) /\ (/) e/ ( fi ` A ) /\ A. x e. ( fi ` A ) A. y e. ( fi ` A ) E. z e. ( fi ` A ) z C_ ( x i^i y ) ) )
35 isfbas2 21639 . . . . 5 |- ( X e. V -> ( ( fi ` A ) e. ( fBas ` X ) <-> ( ( fi ` A ) C_ ~P X /\ ( ( fi ` A ) =/= (/) /\ (/) e/ ( fi ` A ) /\ A. x e. ( fi ` A ) A. y e. ( fi ` A ) E. z e. ( fi ` A ) z C_ ( x i^i y ) ) ) ) )
3635adantr 481 . . . 4 |- ( ( X e. V /\ ( A C_ ~P X /\ A =/= (/) /\ -. (/) e. ( fi ` A ) ) ) -> ( ( fi ` A ) e. ( fBas ` X ) <-> ( ( fi ` A ) C_ ~P X /\ ( ( fi ` A ) =/= (/) /\ (/) e/ ( fi ` A ) /\ A. x e. ( fi ` A ) A. y e. ( fi ` A ) E. z e. ( fi ` A ) z C_ ( x i^i y ) ) ) ) )
3717, 34, 36mpbir2and 957 . . 3 |- ( ( X e. V /\ ( A C_ ~P X /\ A =/= (/) /\ -. (/) e. ( fi ` A ) ) ) -> ( fi ` A ) e. ( fBas ` X ) )
3837ex 450 . 2 |- ( X e. V -> ( ( A C_ ~P X /\ A =/= (/) /\ -. (/) e. ( fi ` A ) ) -> ( fi ` A ) e. ( fBas ` X ) ) )
3914, 38impbid2 216 1 |- ( X e. V -> ( ( fi ` A ) e. ( fBas ` X ) <-> ( A C_ ~P X /\ A =/= (/) /\ -. (/) e. ( fi ` A ) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   e. wcel 1990   =/= wne 2794   e/ wnel 2897   A.wral 2912   E.wrex 2913   _Vcvv 3200   i^i cin 3573   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   ` cfv 5888   ficfi 8316   fBascfbas 19734
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-fin 7959  df-fi 8317  df-fbas 19743
This theorem is referenced by:  isufil2  21712  ufileu  21723  filufint  21724  fmfnfm  21762  hausflim  21785  flimclslem  21788  fclsfnflim  21831  flimfnfcls  21832  fclscmp  21834
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