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Theorem isbasisg 17043
Description: Express the predicate " B is a basis for a topology." (Contributed by NM, 17-Jul-2006.)
Assertion
Ref Expression
isbasisg  |-  ( B  e.  C  ->  ( 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 ) ) ) )
Distinct variable group:    x, y, B
Allowed substitution hints:    C( x, y)

Proof of Theorem isbasisg
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 ineq1 3521 . . . . . 6  |-  ( z  =  B  ->  (
z  i^i  ~P (
x  i^i  y )
)  =  ( B  i^i  ~P ( x  i^i  y ) ) )
21unieqd 4050 . . . . 5  |-  ( z  =  B  ->  U. (
z  i^i  ~P (
x  i^i  y )
)  =  U. ( B  i^i  ~P ( x  i^i  y ) ) )
32sseq2d 3362 . . . 4  |-  ( z  =  B  ->  (
( x  i^i  y
)  C_  U. (
z  i^i  ~P (
x  i^i  y )
)  <->  ( x  i^i  y )  C_  U. ( B  i^i  ~P ( x  i^i  y ) ) ) )
43raleqbi1dv 2918 . . 3  |-  ( z  =  B  ->  ( A. y  e.  z 
( x  i^i  y
)  C_  U. (
z  i^i  ~P (
x  i^i  y )
)  <->  A. y  e.  B  ( x  i^i  y
)  C_  U. ( B  i^i  ~P ( x  i^i  y ) ) ) )
54raleqbi1dv 2918 . 2  |-  ( z  =  B  ->  ( A. x  e.  z  A. y  e.  z 
( x  i^i  y
)  C_  U. (
z  i^i  ~P (
x  i^i  y )
)  <->  A. x  e.  B  A. y  e.  B  ( x  i^i  y
)  C_  U. ( B  i^i  ~P ( x  i^i  y ) ) ) )
6 df-bases 16996 . 2  |-  TopBases  =  {
z  |  A. x  e.  z  A. y  e.  z  ( x  i^i  y )  C_  U. (
z  i^i  ~P (
x  i^i  y )
) }
75, 6elab2g 3090 1  |-  ( B  e.  C  ->  ( 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 ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    = wceq 1653    e. wcel 1727   A.wral 2711    i^i cin 3305    C_ wss 3306   ~Pcpw 3823   U.cuni 4039   TopBasesctb 16993
This theorem is referenced by:  isbasis2g  17044  basis1  17046  basdif0  17049  baspartn  17050  basqtop  17774
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ral 2716  df-rex 2717  df-v 2964  df-in 3313  df-ss 3320  df-uni 4040  df-bases 16996
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