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Definition df-lly 17192
Description: Define a space that is locally  A, where  A is a topological property like "compact", "connected", or "path-connected". A topological space is locally 
A if every neighborhood of a point contains an open sub-neighborhood that is  A in the subspace topology. (Contributed by Mario Carneiro, 2-Mar-2015.)
Assertion
Ref Expression
df-lly  |- Locally  A  =  { j  e.  Top  | 
A. x  e.  j 
A. y  e.  x  E. u  e.  (
j  i^i  ~P x
) ( y  e.  u  /\  ( jt  u )  e.  A ) }
Distinct variable group:    u, j, x, y, A

Detailed syntax breakdown of Definition df-lly
StepHypRef Expression
1 cA . . 3  class  A
21clly 17190 . 2  class Locally  A
3 vy . . . . . . . 8  set  y
4 vu . . . . . . . 8  set  u
53, 4wel 1685 . . . . . . 7  wff  y  e.  u
6 vj . . . . . . . . . 10  set  j
76cv 1622 . . . . . . . . 9  class  j
84cv 1622 . . . . . . . . 9  class  u
9 crest 13325 . . . . . . . . 9  classt
107, 8, 9co 5858 . . . . . . . 8  class  ( jt  u )
1110, 1wcel 1684 . . . . . . 7  wff  ( jt  u )  e.  A
125, 11wa 358 . . . . . 6  wff  ( y  e.  u  /\  (
jt  u )  e.  A
)
13 vx . . . . . . . . 9  set  x
1413cv 1622 . . . . . . . 8  class  x
1514cpw 3625 . . . . . . 7  class  ~P x
167, 15cin 3151 . . . . . 6  class  ( j  i^i  ~P x )
1712, 4, 16wrex 2544 . . . . 5  wff  E. u  e.  ( j  i^i  ~P x ) ( y  e.  u  /\  (
jt  u )  e.  A
)
1817, 3, 14wral 2543 . . . 4  wff  A. y  e.  x  E. u  e.  ( j  i^i  ~P x ) ( y  e.  u  /\  (
jt  u )  e.  A
)
1918, 13, 7wral 2543 . . 3  wff  A. x  e.  j  A. y  e.  x  E. u  e.  ( j  i^i  ~P x ) ( y  e.  u  /\  (
jt  u )  e.  A
)
20 ctop 16631 . . 3  class  Top
2119, 6, 20crab 2547 . 2  class  { j  e.  Top  |  A. x  e.  j  A. y  e.  x  E. u  e.  ( j  i^i  ~P x ) ( y  e.  u  /\  ( jt  u )  e.  A
) }
222, 21wceq 1623 1  wff Locally  A  =  { j  e.  Top  | 
A. x  e.  j 
A. y  e.  x  E. u  e.  (
j  i^i  ~P x
) ( y  e.  u  /\  ( jt  u )  e.  A ) }
Colors of variables: wff set class
This definition is referenced by:  islly  17194  llyeq  17196
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