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Theorem basis1 19578
Description: Property of a basis. (Contributed by NM, 16-Jul-2006.)
Assertion
Ref Expression
basis1  |-  ( ( B  e.  TopBases  /\  C  e.  B  /\  D  e.  B )  ->  ( C  i^i  D )  C_  U. ( B  i^i  ~P ( C  i^i  D ) ) )

Proof of Theorem basis1
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isbasisg 19575 . . . 4  |-  ( B  e.  TopBases  ->  ( B  e.  TopBases  <->  A. x  e.  B  A. y  e.  B  (
x  i^i  y )  C_ 
U. ( B  i^i  ~P ( x  i^i  y
) ) ) )
21ibi 241 . . 3  |-  ( B  e.  TopBases  ->  A. x  e.  B  A. y  e.  B  ( x  i^i  y
)  C_  U. ( B  i^i  ~P ( x  i^i  y ) ) )
3 ineq1 3689 . . . . 5  |-  ( x  =  C  ->  (
x  i^i  y )  =  ( C  i^i  y ) )
43pweqd 4020 . . . . . . 7  |-  ( x  =  C  ->  ~P ( x  i^i  y
)  =  ~P ( C  i^i  y ) )
54ineq2d 3696 . . . . . 6  |-  ( x  =  C  ->  ( B  i^i  ~P ( x  i^i  y ) )  =  ( B  i^i  ~P ( C  i^i  y
) ) )
65unieqd 4261 . . . . 5  |-  ( x  =  C  ->  U. ( B  i^i  ~P ( x  i^i  y ) )  =  U. ( B  i^i  ~P ( C  i^i  y ) ) )
73, 6sseq12d 3528 . . . 4  |-  ( x  =  C  ->  (
( x  i^i  y
)  C_  U. ( B  i^i  ~P ( x  i^i  y ) )  <-> 
( C  i^i  y
)  C_  U. ( B  i^i  ~P ( C  i^i  y ) ) ) )
8 ineq2 3690 . . . . 5  |-  ( y  =  D  ->  ( C  i^i  y )  =  ( C  i^i  D
) )
98pweqd 4020 . . . . . . 7  |-  ( y  =  D  ->  ~P ( C  i^i  y
)  =  ~P ( C  i^i  D ) )
109ineq2d 3696 . . . . . 6  |-  ( y  =  D  ->  ( B  i^i  ~P ( C  i^i  y ) )  =  ( B  i^i  ~P ( C  i^i  D
) ) )
1110unieqd 4261 . . . . 5  |-  ( y  =  D  ->  U. ( B  i^i  ~P ( C  i^i  y ) )  =  U. ( B  i^i  ~P ( C  i^i  D ) ) )
128, 11sseq12d 3528 . . . 4  |-  ( y  =  D  ->  (
( C  i^i  y
)  C_  U. ( B  i^i  ~P ( C  i^i  y ) )  <-> 
( C  i^i  D
)  C_  U. ( B  i^i  ~P ( C  i^i  D ) ) ) )
137, 12rspc2v 3219 . . 3  |-  ( ( C  e.  B  /\  D  e.  B )  ->  ( A. x  e.  B  A. y  e.  B  ( x  i^i  y )  C_  U. ( B  i^i  ~P ( x  i^i  y ) )  ->  ( C  i^i  D )  C_  U. ( B  i^i  ~P ( C  i^i  D ) ) ) )
142, 13syl5com 30 . 2  |-  ( B  e.  TopBases  ->  ( ( C  e.  B  /\  D  e.  B )  ->  ( C  i^i  D )  C_  U. ( B  i^i  ~P ( C  i^i  D ) ) ) )
15143impib 1194 1  |-  ( ( B  e.  TopBases  /\  C  e.  B  /\  D  e.  B )  ->  ( C  i^i  D )  C_  U. ( B  i^i  ~P ( C  i^i  D ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819   A.wral 2807    i^i cin 3470    C_ wss 3471   ~Pcpw 4015   U.cuni 4251   TopBasesctb 19525
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ral 2812  df-rex 2813  df-v 3111  df-in 3478  df-ss 3485  df-pw 4017  df-uni 4252  df-bases 19528
This theorem is referenced by: (None)
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