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Theorem islly 17210
Description: The property of being a locally  A topological space. (Contributed by Mario Carneiro, 2-Mar-2015.)
Assertion
Ref Expression
islly  |-  ( J  e. 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 groups:    x, u, y, A    u, J, x, y

Proof of Theorem islly
Dummy variable  j is distinct from all other variables.
StepHypRef Expression
1 ineq1 3376 . . . . 5  |-  ( j  =  J  ->  (
j  i^i  ~P x
)  =  ( J  i^i  ~P x ) )
2 oveq1 5881 . . . . . . 7  |-  ( j  =  J  ->  (
jt  u )  =  ( Jt  u ) )
32eleq1d 2362 . . . . . 6  |-  ( j  =  J  ->  (
( jt  u )  e.  A  <->  ( Jt  u )  e.  A
) )
43anbi2d 684 . . . . 5  |-  ( j  =  J  ->  (
( y  e.  u  /\  ( jt  u )  e.  A
)  <->  ( y  e.  u  /\  ( Jt  u )  e.  A ) ) )
51, 4rexeqbidv 2762 . . . 4  |-  ( j  =  J  ->  ( E. u  e.  (
j  i^i  ~P x
) ( y  e.  u  /\  ( jt  u )  e.  A )  <->  E. u  e.  ( J  i^i  ~P x ) ( y  e.  u  /\  ( Jt  u )  e.  A
) ) )
65ralbidv 2576 . . 3  |-  ( j  =  J  ->  ( A. y  e.  x  E. u  e.  (
j  i^i  ~P x
) ( y  e.  u  /\  ( jt  u )  e.  A )  <->  A. y  e.  x  E. u  e.  ( J  i^i  ~P x ) ( y  e.  u  /\  ( Jt  u )  e.  A
) ) )
76raleqbi1dv 2757 . 2  |-  ( j  =  J  ->  ( A. x  e.  j  A. y  e.  x  E. u  e.  (
j  i^i  ~P x
) ( y  e.  u  /\  ( jt  u )  e.  A )  <->  A. x  e.  J  A. y  e.  x  E. u  e.  ( J  i^i  ~P x ) ( y  e.  u  /\  ( Jt  u )  e.  A
) ) )
8 df-lly 17208 . 2  |- 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 ) }
97, 8elrab2 2938 1  |-  ( J  e. 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
Syntax hints:    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   E.wrex 2557    i^i cin 3164   ~Pcpw 3638  (class class class)co 5874   ↾t crest 13341   Topctop 16647  Locally clly 17206
This theorem is referenced by:  llytop  17214  llyi  17216  llyss  17221  subislly  17223  restnlly  17224  restlly  17225  islly2  17226  llyrest  17227  llyidm  17230  dislly  17239  txlly  17346  cnllyscon  23791  rellyscon  23797
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-iota 5235  df-fv 5279  df-ov 5877  df-lly 17208
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