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Theorem isbasisg 16971
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 3499 . . . . . 6  |-  ( z  =  B  ->  (
z  i^i  ~P (
x  i^i  y )
)  =  ( B  i^i  ~P ( x  i^i  y ) ) )
21unieqd 3990 . . . . 5  |-  ( z  =  B  ->  U. (
z  i^i  ~P (
x  i^i  y )
)  =  U. ( B  i^i  ~P ( x  i^i  y ) ) )
32sseq2d 3340 . . . 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 2876 . . 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 2876 . 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 16924 . 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 3048 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 177    = wceq 1649    e. wcel 1721   A.wral 2670    i^i cin 3283    C_ wss 3284   ~Pcpw 3763   U.cuni 3979   TopBasesctb 16921
This theorem is referenced by:  isbasis2g  16972  basis1  16974  basdif0  16977  baspartn  16978  basqtop  17700
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ral 2675  df-rex 2676  df-v 2922  df-in 3291  df-ss 3298  df-uni 3980  df-bases 16924
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