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Theorem isbasisg 19533
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 3607 . . . . . 6  |-  ( z  =  B  ->  (
z  i^i  ~P (
x  i^i  y )
)  =  ( B  i^i  ~P ( x  i^i  y ) ) )
21unieqd 4173 . . . . 5  |-  ( z  =  B  ->  U. (
z  i^i  ~P (
x  i^i  y )
)  =  U. ( B  i^i  ~P ( x  i^i  y ) ) )
32sseq2d 3445 . . . 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 2987 . . 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 2987 . 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 19486 . 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 3173 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1399    e. wcel 1826   A.wral 2732    i^i cin 3388    C_ wss 3389   ~Pcpw 3927   U.cuni 4163   TopBasesctb 19483
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360
This theorem depends on definitions:  df-bi 185  df-an 369  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ral 2737  df-rex 2738  df-v 3036  df-in 3396  df-ss 3403  df-uni 4164  df-bases 19486
This theorem is referenced by:  isbasis2g  19534  basis1  19536  basdif0  19539  baspartn  19540  basqtop  20297
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