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Definition df-bases 16638
Description: Define the class of all topological bases. Equivalent to definition of basis in [Munkres] p. 78 (see isbasis2g 16686). Note that "bases" is the plural of "basis." (Contributed by NM, 17-Jul-2006.)
Assertion
Ref Expression
df-bases  |-  TopBases  =  {
x  |  A. y  e.  x  A. z  e.  x  ( y  i^i  z )  C_  U. (
x  i^i  ~P (
y  i^i  z )
) }
Distinct variable group:    x, y, z

Detailed syntax breakdown of Definition df-bases
StepHypRef Expression
1 ctb 16635 . 2  class  TopBases
2 vy . . . . . . . 8  set  y
32cv 1622 . . . . . . 7  class  y
4 vz . . . . . . . 8  set  z
54cv 1622 . . . . . . 7  class  z
63, 5cin 3151 . . . . . 6  class  ( y  i^i  z )
7 vx . . . . . . . . 9  set  x
87cv 1622 . . . . . . . 8  class  x
96cpw 3625 . . . . . . . 8  class  ~P (
y  i^i  z )
108, 9cin 3151 . . . . . . 7  class  ( x  i^i  ~P ( y  i^i  z ) )
1110cuni 3827 . . . . . 6  class  U. (
x  i^i  ~P (
y  i^i  z )
)
126, 11wss 3152 . . . . 5  wff  ( y  i^i  z )  C_  U. ( x  i^i  ~P ( y  i^i  z
) )
1312, 4, 8wral 2543 . . . 4  wff  A. z  e.  x  ( y  i^i  z )  C_  U. (
x  i^i  ~P (
y  i^i  z )
)
1413, 2, 8wral 2543 . . 3  wff  A. y  e.  x  A. z  e.  x  ( y  i^i  z )  C_  U. (
x  i^i  ~P (
y  i^i  z )
)
1514, 7cab 2269 . 2  class  { x  |  A. y  e.  x  A. z  e.  x  ( y  i^i  z
)  C_  U. (
x  i^i  ~P (
y  i^i  z )
) }
161, 15wceq 1623 1  wff  TopBases  =  {
x  |  A. y  e.  x  A. z  e.  x  ( y  i^i  z )  C_  U. (
x  i^i  ~P (
y  i^i  z )
) }
Colors of variables: wff set class
This definition is referenced by:  isbasisg  16685
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