Theorem fbasweak 21669

Description: A filter base on any set is also a filter base on any larger set. (Contributed by Stefan O'Rear, 2-Aug-2015.)

Assertion

Ref	Expression
fbasweak	|- ( ( F e. ( fBas ` X ) /\ F C_ ~P Y /\ Y e. V ) -> F e. ( fBas ` Y ) )

Proof of Theorem fbasweak

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp2 1062	. 2 |- ( ( F e. ( fBas ` X ) /\ F C_ ~P Y /\ Y e. V ) -> F C_ ~P Y )
2		simp1 1061	. . . 4 |- ( ( F e. ( fBas ` X ) /\ F C_ ~P Y /\ Y e. V ) -> F e. ( fBas ` X ) )
3		elfvdm 6220	. . . . . 6 |- ( F e. ( fBas ` X ) -> X e. dom fBas )
4	3	3ad2ant1 1082	. . . . 5 |- ( ( F e. ( fBas ` X ) /\ F C_ ~P Y /\ Y e. V ) -> X e. dom fBas )
5		isfbas 21633	. . . . 5 |- ( X e. dom fBas -> ( F e. ( fBas ` X ) <-> ( F C_ ~P X /\ ( F =/= (/) /\ (/) e/ F /\ A. x e. F A. y e. F ( F i^i ~P ( x i^i y ) ) =/= (/) ) ) ) )
6	4, 5	syl 17	. . . 4 |- ( ( F e. ( fBas ` X ) /\ F C_ ~P Y /\ Y e. V ) -> ( F e. ( fBas ` X ) <-> ( F C_ ~P X /\ ( F =/= (/) /\ (/) e/ F /\ A. x e. F A. y e. F ( F i^i ~P ( x i^i y ) ) =/= (/) ) ) ) )
7	2, 6	mpbid 222	. . 3 |- ( ( F e. ( fBas ` X ) /\ F C_ ~P Y /\ Y e. V ) -> ( F C_ ~P X /\ ( F =/= (/) /\ (/) e/ F /\ A. x e. F A. y e. F ( F i^i ~P ( x i^i y ) ) =/= (/) ) ) )
8	7	simprd 479	. 2 |- ( ( F e. ( fBas ` X ) /\ F C_ ~P Y /\ Y e. V ) -> ( F =/= (/) /\ (/) e/ F /\ A. x e. F A. y e. F ( F i^i ~P ( x i^i y ) ) =/= (/) ) )
9		isfbas 21633	. . 3 |- ( Y e. V -> ( F e. ( fBas ` Y ) <-> ( F C_ ~P Y /\ ( F =/= (/) /\ (/) e/ F /\ A. x e. F A. y e. F ( F i^i ~P ( x i^i y ) ) =/= (/) ) ) ) )
10	9	3ad2ant3 1084	. 2 |- ( ( F e. ( fBas ` X ) /\ F C_ ~P Y /\ Y e. V ) -> ( F e. ( fBas ` Y ) <-> ( F C_ ~P Y /\ ( F =/= (/) /\ (/) e/ F /\ A. x e. F A. y e. F ( F i^i ~P ( x i^i y ) ) =/= (/) ) ) ) )
11	1, 8, 10	mpbir2and 957	1 |- ( ( F e. ( fBas ` X ) /\ F C_ ~P Y /\ Y e. V ) -> F e. ( fBas ` Y ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   e. wcel 1990   =/= wne 2794   e/ wnel 2897   A.wral 2912   i^i cin 3573   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   dom cdm 5114   ` cfv 5888   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

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-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-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-fv 5896  df-fbas 19743

This theorem is referenced by:  snfbas  21670  fgabs  21683  fgtr  21694  trfg  21695  ssufl  21722  cfiluweak  22099  cfilresi  23093  cmetss  23113  minveclem4a  23201  minveclem4  23203