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Theorem fbasssin 20065
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 5883 . . . . . . 7  |-  ( F  e.  ( fBas `  X
)  ->  X  e.  dom  fBas )
2 isfbas2 20064 . . . . . . 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 241 . . . . 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 463 . . . 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 1005 . . 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 3686 . . . . . 6  |-  ( y  =  A  ->  (
y  i^i  z )  =  ( A  i^i  z ) )
87sseq2d 3525 . . . . 5  |-  ( y  =  A  ->  (
x  C_  ( y  i^i  z )  <->  x  C_  ( A  i^i  z ) ) )
98rexbidv 2966 . . . 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 3687 . . . . . 6  |-  ( z  =  B  ->  ( A  i^i  z )  =  ( A  i^i  B
) )
1110sseq2d 3525 . . . . 5  |-  ( z  =  B  ->  (
x  C_  ( A  i^i  z )  <->  x  C_  ( A  i^i  B ) ) )
1211rexbidv 2966 . . . 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 3216 . . 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 30 . 2  |-  ( F  e.  ( fBas `  X
)  ->  ( ( A  e.  F  /\  B  e.  F )  ->  E. x  e.  F  x  C_  ( A  i^i  B ) ) )
15143impib 1189 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762    =/= wne 2655    e/ wnel 2656   A.wral 2807   E.wrex 2808    i^i cin 3468    C_ wss 3469   (/)c0 3778   ~Pcpw 4003   dom cdm 4992   ` cfv 5579   fBascfbas 18170
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fv 5587  df-fbas 18180
This theorem is referenced by:  fbssfi  20066  fbncp  20068  fbun  20069  fbfinnfr  20070  trfbas2  20072  filin  20083  fgcl  20107  fbasrn  20113
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