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Theorem cfiluweak 22099
Description: A Cauchy filter base is also a Cauchy filter base on any coarser uniform structure. (Contributed by Thierry Arnoux, 24-Jan-2018.)
Assertion
Ref Expression
cfiluweak |- ( ( U e. (UnifOn ` X ) /\ A C_ X /\ F e. (CauFilu ` ( U ↾t ( A X. A ) ) ) ) -> F e. (CauFilu ` U ) )
Proof of Theorem cfiluweak
Dummy variables u a v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 trust 22033 . . . . 5 |- ( ( U e. (UnifOn ` X ) /\ A C_ X ) -> ( U ↾t ( A X. A ) ) e. (UnifOn ` A ) )
2 iscfilu 22092 . . . . . 6 |- ( ( U ↾t ( A X. A ) ) e. (UnifOn ` A ) -> ( F e. (CauFilu ` ( U ↾t ( A X. A ) ) ) <-> ( F e. ( fBas ` A ) /\ A. u e. ( U ↾t ( A X. A ) ) E. a e. F ( a X. a ) C_ u ) ) )
32biimpa 501 . . . . 5 |- ( ( ( U ↾t ( A X. A ) ) e. (UnifOn ` A ) /\ F e. (CauFilu ` ( U ↾t ( A X. A ) ) ) ) -> ( F e. ( fBas ` A ) /\ A. u e. ( U ↾t ( A X. A ) ) E. a e. F ( a X. a ) C_ u ) )
41, 3stoic3 1701 . . . 4 |- ( ( U e. (UnifOn ` X ) /\ A C_ X /\ F e. (CauFilu ` ( U ↾t ( A X. A ) ) ) ) -> ( F e. ( fBas ` A ) /\ A. u e. ( U ↾t ( A X. A ) ) E. a e. F ( a X. a ) C_ u ) )
54simpld 475 . . 3 |- ( ( U e. (UnifOn ` X ) /\ A C_ X /\ F e. (CauFilu ` ( U ↾t ( A X. A ) ) ) ) -> F e. ( fBas ` A ) )
6 fbsspw 21636 . . . . 5 |- ( F e. ( fBas ` A ) -> F C_ ~P A )
75, 6syl 17 . . . 4 |- ( ( U e. (UnifOn ` X ) /\ A C_ X /\ F e. (CauFilu ` ( U ↾t ( A X. A ) ) ) ) -> F C_ ~P A )
8 simp2 1062 . . . . 5 |- ( ( U e. (UnifOn ` X ) /\ A C_ X /\ F e. (CauFilu ` ( U ↾t ( A X. A ) ) ) ) -> A C_ X )
9 sspwb 4917 . . . . 5 |- ( A C_ X <-> ~P A C_ ~P X )
108, 9sylib 208 . . . 4 |- ( ( U e. (UnifOn ` X ) /\ A C_ X /\ F e. (CauFilu ` ( U ↾t ( A X. A ) ) ) ) -> ~P A C_ ~P X )
117, 10sstrd 3613 . . 3 |- ( ( U e. (UnifOn ` X ) /\ A C_ X /\ F e. (CauFilu ` ( U ↾t ( A X. A ) ) ) ) -> F C_ ~P X )
12 simp1 1061 . . . 4 |- ( ( U e. (UnifOn ` X ) /\ A C_ X /\ F e. (CauFilu ` ( U ↾t ( A X. A ) ) ) ) -> U e. (UnifOn ` X ) )
1312elfvexd 6222 . . 3 |- ( ( U e. (UnifOn ` X ) /\ A C_ X /\ F e. (CauFilu ` ( U ↾t ( A X. A ) ) ) ) -> X e. _V )
14 fbasweak 21669 . . 3 |- ( ( F e. ( fBas ` A ) /\ F C_ ~P X /\ X e. _V ) -> F e. ( fBas ` X ) )
155, 11, 13, 14syl3anc 1326 . 2 |- ( ( U e. (UnifOn ` X ) /\ A C_ X /\ F e. (CauFilu ` ( U ↾t ( A X. A ) ) ) ) -> F e. ( fBas ` X ) )
1612adantr 481 . . . . . 6 |- ( ( ( U e. (UnifOn ` X ) /\ A C_ X /\ F e. (CauFilu ` ( U ↾t ( A X. A ) ) ) ) /\ v e. U ) -> U e. (UnifOn ` X ) )
1713adantr 481 . . . . . . . 8 |- ( ( ( U e. (UnifOn ` X ) /\ A C_ X /\ F e. (CauFilu ` ( U ↾t ( A X. A ) ) ) ) /\ v e. U ) -> X e. _V )
188adantr 481 . . . . . . . 8 |- ( ( ( U e. (UnifOn ` X ) /\ A C_ X /\ F e. (CauFilu ` ( U ↾t ( A X. A ) ) ) ) /\ v e. U ) -> A C_ X )
1917, 18ssexd 4805 . . . . . . 7 |- ( ( ( U e. (UnifOn ` X ) /\ A C_ X /\ F e. (CauFilu ` ( U ↾t ( A X. A ) ) ) ) /\ v e. U ) -> A e. _V )
20 xpexg 6960 . . . . . . 7 |- ( ( A e. _V /\ A e. _V ) -> ( A X. A ) e. _V )
2119, 19, 20syl2anc 693 . . . . . 6 |- ( ( ( U e. (UnifOn ` X ) /\ A C_ X /\ F e. (CauFilu ` ( U ↾t ( A X. A ) ) ) ) /\ v e. U ) -> ( A X. A ) e. _V )
22 simpr 477 . . . . . 6 |- ( ( ( U e. (UnifOn ` X ) /\ A C_ X /\ F e. (CauFilu ` ( U ↾t ( A X. A ) ) ) ) /\ v e. U ) -> v e. U )
23 elrestr 16089 . . . . . 6 |- ( ( U e. (UnifOn ` X ) /\ ( A X. A ) e. _V /\ v e. U ) -> ( v i^i ( A X. A ) ) e. ( U ↾t ( A X. A ) ) )
2416, 21, 22, 23syl3anc 1326 . . . . 5 |- ( ( ( U e. (UnifOn ` X ) /\ A C_ X /\ F e. (CauFilu ` ( U ↾t ( A X. A ) ) ) ) /\ v e. U ) -> ( v i^i ( A X. A ) ) e. ( U ↾t ( A X. A ) ) )
254simprd 479 . . . . . 6 |- ( ( U e. (UnifOn ` X ) /\ A C_ X /\ F e. (CauFilu ` ( U ↾t ( A X. A ) ) ) ) -> A. u e. ( U ↾t ( A X. A ) ) E. a e. F ( a X. a ) C_ u )
2625adantr 481 . . . . 5 |- ( ( ( U e. (UnifOn ` X ) /\ A C_ X /\ F e. (CauFilu ` ( U ↾t ( A X. A ) ) ) ) /\ v e. U ) -> A. u e. ( U ↾t ( A X. A ) ) E. a e. F ( a X. a ) C_ u )
27 sseq2 3627 . . . . . . 7 |- ( u = ( v i^i ( A X. A ) ) -> ( ( a X. a ) C_ u <-> ( a X. a ) C_ ( v i^i ( A X. A ) ) ) )
2827rexbidv 3052 . . . . . 6 |- ( u = ( v i^i ( A X. A ) ) -> ( E. a e. F ( a X. a ) C_ u <-> E. a e. F ( a X. a ) C_ ( v i^i ( A X. A ) ) ) )
2928rspcva 3307 . . . . 5 |- ( ( ( v i^i ( A X. A ) ) e. ( U ↾t ( A X. A ) ) /\ A. u e. ( U ↾t ( A X. A ) ) E. a e. F ( a X. a ) C_ u ) -> E. a e. F ( a X. a ) C_ ( v i^i ( A X. A ) ) )
3024, 26, 29syl2anc 693 . . . 4 |- ( ( ( U e. (UnifOn ` X ) /\ A C_ X /\ F e. (CauFilu ` ( U ↾t ( A X. A ) ) ) ) /\ v e. U ) -> E. a e. F ( a X. a ) C_ ( v i^i ( A X. A ) ) )
31 inss1 3833 . . . . . 6 |- ( v i^i ( A X. A ) ) C_ v
32 sstr 3611 . . . . . 6 |- ( ( ( a X. a ) C_ ( v i^i ( A X. A ) ) /\ ( v i^i ( A X. A ) ) C_ v ) -> ( a X. a ) C_ v )
3331, 32mpan2 707 . . . . 5 |- ( ( a X. a ) C_ ( v i^i ( A X. A ) ) -> ( a X. a ) C_ v )
3433reximi 3011 . . . 4 |- ( E. a e. F ( a X. a ) C_ ( v i^i ( A X. A ) ) -> E. a e. F ( a X. a ) C_ v )
3530, 34syl 17 . . 3 |- ( ( ( U e. (UnifOn ` X ) /\ A C_ X /\ F e. (CauFilu ` ( U ↾t ( A X. A ) ) ) ) /\ v e. U ) -> E. a e. F ( a X. a ) C_ v )
3635ralrimiva 2966 . 2 |- ( ( U e. (UnifOn ` X ) /\ A C_ X /\ F e. (CauFilu ` ( U ↾t ( A X. A ) ) ) ) -> A. v e. U E. a e. F ( a X. a ) C_ v )
37 iscfilu 22092 . . 3 |- ( U e. (UnifOn ` X ) -> ( F e. (CauFilu ` U ) <-> ( F e. ( fBas ` X ) /\ A. v e. U E. a e. F ( a X. a ) C_ v ) ) )
38373ad2ant1 1082 . 2 |- ( ( U e. (UnifOn ` X ) /\ A C_ X /\ F e. (CauFilu ` ( U ↾t ( A X. A ) ) ) ) -> ( F e. (CauFilu ` U ) <-> ( F e. ( fBas ` X ) /\ A. v e. U E. a e. F ( a X. a ) C_ v ) ) )
3915, 36, 38mpbir2and 957 1 |- ( ( U e. (UnifOn ` X ) /\ A C_ X /\ F e. (CauFilu ` ( U ↾t ( A X. A ) ) ) ) -> F e. (CauFilu ` U ) )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   _Vcvv 3200   i^i cin 3573   C_ wss 3574   ~Pcpw 4158   X. cxp 5112   ` cfv 5888  (class class class)co 6650   ↾t crest 16081   fBascfbas 19734  UnifOncust 22003  CauFiluccfilu 22090
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-rep 4771  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-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-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  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-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-1st 7168  df-2nd 7169  df-rest 16083  df-fbas 19743  df-ust 22004  df-cfilu 22091
This theorem is referenced by: (None)
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