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Theorem fbasssin 17868
Description: A filter base contains subsets of its pairwise intersections. (Contributed by Jeff Hankins, 1-Sep-2009.) (Revised by Jeff Hankins, 1-Dec-2010.)
Assertion
Ref Expression
fbasssin  |-  ( ( F  e.  ( fBas `  X )  /\  A  e.  F  /\  B  e.  F )  ->  E. x  e.  F  x  C_  ( A  i^i  B ) )
Distinct variable groups:    x, A    x, B    x, F    x, X

Proof of Theorem fbasssin
Dummy variables  y 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elfvdm 5757 . . . . . . 7  |-  ( F  e.  ( fBas `  X
)  ->  X  e.  dom  fBas )
2 isfbas2 17867 . . . . . . 7  |-  ( X  e.  dom  fBas  ->  ( F  e.  ( fBas `  X )  <->  ( F  C_ 
~P X  /\  ( F  =/=  (/)  /\  (/)  e/  F  /\  A. y  e.  F  A. z  e.  F  E. x  e.  F  x  C_  ( y  i^i  z ) ) ) ) )
31, 2syl 16 . . . . . 6  |-  ( F  e.  ( fBas `  X
)  ->  ( F  e.  ( fBas `  X
)  <->  ( F  C_  ~P X  /\  ( F  =/=  (/)  /\  (/)  e/  F  /\  A. y  e.  F  A. z  e.  F  E. x  e.  F  x  C_  ( y  i^i  z ) ) ) ) )
43ibi 233 . . . . 5  |-  ( F  e.  ( fBas `  X
)  ->  ( F  C_ 
~P X  /\  ( F  =/=  (/)  /\  (/)  e/  F  /\  A. y  e.  F  A. z  e.  F  E. x  e.  F  x  C_  ( y  i^i  z ) ) ) )
54simprd 450 . . . 4  |-  ( F  e.  ( fBas `  X
)  ->  ( F  =/=  (/)  /\  (/)  e/  F  /\  A. y  e.  F  A. z  e.  F  E. x  e.  F  x  C_  ( y  i^i  z ) ) )
65simp3d 971 . . 3  |-  ( F  e.  ( fBas `  X
)  ->  A. y  e.  F  A. z  e.  F  E. x  e.  F  x  C_  (
y  i^i  z )
)
7 ineq1 3535 . . . . . 6  |-  ( y  =  A  ->  (
y  i^i  z )  =  ( A  i^i  z ) )
87sseq2d 3376 . . . . 5  |-  ( y  =  A  ->  (
x  C_  ( y  i^i  z )  <->  x  C_  ( A  i^i  z ) ) )
98rexbidv 2726 . . . 4  |-  ( y  =  A  ->  ( E. x  e.  F  x  C_  ( y  i^i  z )  <->  E. x  e.  F  x  C_  ( A  i^i  z ) ) )
10 ineq2 3536 . . . . . 6  |-  ( z  =  B  ->  ( A  i^i  z )  =  ( A  i^i  B
) )
1110sseq2d 3376 . . . . 5  |-  ( z  =  B  ->  (
x  C_  ( A  i^i  z )  <->  x  C_  ( A  i^i  B ) ) )
1211rexbidv 2726 . . . 4  |-  ( z  =  B  ->  ( E. x  e.  F  x  C_  ( A  i^i  z )  <->  E. x  e.  F  x  C_  ( A  i^i  B ) ) )
139, 12rspc2v 3058 . . 3  |-  ( ( A  e.  F  /\  B  e.  F )  ->  ( A. y  e.  F  A. z  e.  F  E. x  e.  F  x  C_  (
y  i^i  z )  ->  E. x  e.  F  x  C_  ( A  i^i  B ) ) )
146, 13syl5com 28 . 2  |-  ( F  e.  ( fBas `  X
)  ->  ( ( A  e.  F  /\  B  e.  F )  ->  E. x  e.  F  x  C_  ( A  i^i  B ) ) )
15143impib 1151 1  |-  ( ( F  e.  ( fBas `  X )  /\  A  e.  F  /\  B  e.  F )  ->  E. x  e.  F  x  C_  ( A  i^i  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2599    e/ wnel 2600   A.wral 2705   E.wrex 2706    i^i cin 3319    C_ wss 3320   (/)c0 3628   ~Pcpw 3799   dom cdm 4878   ` cfv 5454   fBascfbas 16689
This theorem is referenced by:  fbssfi  17869  fbncp  17871  fbun  17872  fbfinnfr  17873  trfbas2  17875  filin  17886  fgcl  17910  fbasrn  17916
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fv 5462  df-fbas 16699
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