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Theorem restlly 19099
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 3368 . . . 4  |-  ( (
ph  /\  j  e.  A )  ->  j  e.  Top )
3 simprl 755 . . . . . . 7  |-  ( ( ( ph  /\  j  e.  A )  /\  (
x  e.  j  /\  y  e.  x )
)  ->  x  e.  j )
4 vex 2987 . . . . . . . . 9  |-  x  e. 
_V
54pwid 3886 . . . . . . . 8  |-  x  e. 
~P x
65a1i 11 . . . . . . 7  |-  ( ( ( ph  /\  j  e.  A )  /\  (
x  e.  j  /\  y  e.  x )
)  ->  x  e.  ~P x )
73, 6elind 3552 . . . . . 6  |-  ( ( ( ph  /\  j  e.  A )  /\  (
x  e.  j  /\  y  e.  x )
)  ->  x  e.  ( j  i^i  ~P x ) )
8 simprr 756 . . . . . 6  |-  ( ( ( ph  /\  j  e.  A )  /\  (
x  e.  j  /\  y  e.  x )
)  ->  y  e.  x )
9 restlly.1 . . . . . . . 8  |-  ( (
ph  /\  ( j  e.  A  /\  x  e.  j ) )  -> 
( jt  x )  e.  A
)
109anassrs 648 . . . . . . 7  |-  ( ( ( ph  /\  j  e.  A )  /\  x  e.  j )  ->  (
jt  x )  e.  A
)
1110adantrr 716 . . . . . 6  |-  ( ( ( ph  /\  j  e.  A )  /\  (
x  e.  j  /\  y  e.  x )
)  ->  ( jt  x
)  e.  A )
12 eleq2 2504 . . . . . . . 8  |-  ( u  =  x  ->  (
y  e.  u  <->  y  e.  x ) )
13 oveq2 6111 . . . . . . . . 9  |-  ( u  =  x  ->  (
jt  u )  =  ( jt  x ) )
1413eleq1d 2509 . . . . . . . 8  |-  ( u  =  x  ->  (
( jt  u )  e.  A  <->  ( jt  x )  e.  A
) )
1512, 14anbi12d 710 . . . . . . 7  |-  ( u  =  x  ->  (
( y  e.  u  /\  ( jt  u )  e.  A
)  <->  ( y  e.  x  /\  ( jt  x )  e.  A ) ) )
1615rspcev 3085 . . . . . 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
) )
177, 8, 11, 16syl12anc 1216 . . . . 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
) )
1817ralrimivva 2820 . . . 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
) )
19 islly 19084 . . . 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
) ) )
202, 18, 19sylanbrc 664 . . 3  |-  ( (
ph  /\  j  e.  A )  ->  j  e. Locally  A )
2120ex 434 . 2  |-  ( ph  ->  ( j  e.  A  ->  j  e. Locally  A )
)
2221ssrdv 3374 1  |-  ( ph  ->  A  C_ Locally  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    e. wcel 1756   A.wral 2727   E.wrex 2728    i^i cin 3339    C_ wss 3340   ~Pcpw 3872  (class class class)co 6103   ↾t crest 14371   Topctop 18510  Locally clly 19080
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ral 2732  df-rex 2733  df-rab 2736  df-v 2986  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-op 3896  df-uni 4104  df-br 4305  df-iota 5393  df-fv 5438  df-ov 6106  df-lly 19082
This theorem is referenced by:  llyidm  19104  nllyidm  19105  toplly  19106  hauslly  19108  lly1stc  19112
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