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Theorem restlly 17460
Description: If the property  A passes to open subspaces, then a space which is  A is also locally  A. (Contributed by Mario Carneiro, 2-Mar-2015.)
Hypotheses
Ref Expression
restlly.1  |-  ( (
ph  /\  ( j  e.  A  /\  x  e.  j ) )  -> 
( jt  x )  e.  A
)
restlly.2  |-  ( ph  ->  A  C_  Top )
Assertion
Ref Expression
restlly  |-  ( ph  ->  A  C_ Locally  A )
Distinct variable groups:    x, j, A    ph, j, x

Proof of Theorem restlly
Dummy variables  u  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 restlly.2 . . . . 5  |-  ( ph  ->  A  C_  Top )
21sselda 3284 . . . 4  |-  ( (
ph  /\  j  e.  A )  ->  j  e.  Top )
3 simprl 733 . . . . . . 7  |-  ( ( ( ph  /\  j  e.  A )  /\  (
x  e.  j  /\  y  e.  x )
)  ->  x  e.  j )
4 vex 2895 . . . . . . . . 9  |-  x  e. 
_V
54pwid 3748 . . . . . . . 8  |-  x  e. 
~P x
65a1i 11 . . . . . . 7  |-  ( ( ( ph  /\  j  e.  A )  /\  (
x  e.  j  /\  y  e.  x )
)  ->  x  e.  ~P x )
7 elin 3466 . . . . . . 7  |-  ( x  e.  ( j  i^i 
~P x )  <->  ( x  e.  j  /\  x  e.  ~P x ) )
83, 6, 7sylanbrc 646 . . . . . 6  |-  ( ( ( ph  /\  j  e.  A )  /\  (
x  e.  j  /\  y  e.  x )
)  ->  x  e.  ( j  i^i  ~P x ) )
9 simprr 734 . . . . . 6  |-  ( ( ( ph  /\  j  e.  A )  /\  (
x  e.  j  /\  y  e.  x )
)  ->  y  e.  x )
10 restlly.1 . . . . . . . 8  |-  ( (
ph  /\  ( j  e.  A  /\  x  e.  j ) )  -> 
( jt  x )  e.  A
)
1110anassrs 630 . . . . . . 7  |-  ( ( ( ph  /\  j  e.  A )  /\  x  e.  j )  ->  (
jt  x )  e.  A
)
1211adantrr 698 . . . . . 6  |-  ( ( ( ph  /\  j  e.  A )  /\  (
x  e.  j  /\  y  e.  x )
)  ->  ( jt  x
)  e.  A )
13 eleq2 2441 . . . . . . . 8  |-  ( u  =  x  ->  (
y  e.  u  <->  y  e.  x ) )
14 oveq2 6021 . . . . . . . . 9  |-  ( u  =  x  ->  (
jt  u )  =  ( jt  x ) )
1514eleq1d 2446 . . . . . . . 8  |-  ( u  =  x  ->  (
( jt  u )  e.  A  <->  ( jt  x )  e.  A
) )
1613, 15anbi12d 692 . . . . . . 7  |-  ( u  =  x  ->  (
( y  e.  u  /\  ( jt  u )  e.  A
)  <->  ( y  e.  x  /\  ( jt  x )  e.  A ) ) )
1716rspcev 2988 . . . . . 6  |-  ( ( x  e.  ( j  i^i  ~P x )  /\  ( y  e.  x  /\  ( jt  x )  e.  A ) )  ->  E. u  e.  ( j  i^i  ~P x ) ( y  e.  u  /\  (
jt  u )  e.  A
) )
188, 9, 12, 17syl12anc 1182 . . . . 5  |-  ( ( ( ph  /\  j  e.  A )  /\  (
x  e.  j  /\  y  e.  x )
)  ->  E. u  e.  ( j  i^i  ~P x ) ( y  e.  u  /\  (
jt  u )  e.  A
) )
1918ralrimivva 2734 . . . 4  |-  ( (
ph  /\  j  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
) )
20 islly 17445 . . . 4  |-  ( 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
) ) )
212, 19, 20sylanbrc 646 . . 3  |-  ( (
ph  /\  j  e.  A )  ->  j  e. Locally  A )
2221ex 424 . 2  |-  ( ph  ->  ( j  e.  A  ->  j  e. Locally  A )
)
2322ssrdv 3290 1  |-  ( ph  ->  A  C_ Locally  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    e. wcel 1717   A.wral 2642   E.wrex 2643    i^i cin 3255    C_ wss 3256   ~Pcpw 3735  (class class class)co 6013   ↾t crest 13568   Topctop 16874  Locally clly 17441
This theorem is referenced by:  llyidm  17465  nllyidm  17466  toplly  17467  hauslly  17469  lly1stc  17473
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 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-br 4147  df-iota 5351  df-fv 5395  df-ov 6016  df-lly 17443
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