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Theorem restlly 17551
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 3350 . . . 4  |-  ( (
ph  /\  j  e.  A )  ->  j  e.  Top )
3 simprl 734 . . . . . . 7  |-  ( ( ( ph  /\  j  e.  A )  /\  (
x  e.  j  /\  y  e.  x )
)  ->  x  e.  j )
4 vex 2961 . . . . . . . . 9  |-  x  e. 
_V
54pwid 3814 . . . . . . . 8  |-  x  e. 
~P x
65a1i 11 . . . . . . 7  |-  ( ( ( ph  /\  j  e.  A )  /\  (
x  e.  j  /\  y  e.  x )
)  ->  x  e.  ~P x )
7 elin 3532 . . . . . . 7  |-  ( x  e.  ( j  i^i 
~P x )  <->  ( x  e.  j  /\  x  e.  ~P x ) )
83, 6, 7sylanbrc 647 . . . . . 6  |-  ( ( ( ph  /\  j  e.  A )  /\  (
x  e.  j  /\  y  e.  x )
)  ->  x  e.  ( j  i^i  ~P x ) )
9 simprr 735 . . . . . 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 631 . . . . . . 7  |-  ( ( ( ph  /\  j  e.  A )  /\  x  e.  j )  ->  (
jt  x )  e.  A
)
1211adantrr 699 . . . . . 6  |-  ( ( ( ph  /\  j  e.  A )  /\  (
x  e.  j  /\  y  e.  x )
)  ->  ( jt  x
)  e.  A )
13 eleq2 2499 . . . . . . . 8  |-  ( u  =  x  ->  (
y  e.  u  <->  y  e.  x ) )
14 oveq2 6092 . . . . . . . . 9  |-  ( u  =  x  ->  (
jt  u )  =  ( jt  x ) )
1514eleq1d 2504 . . . . . . . 8  |-  ( u  =  x  ->  (
( jt  u )  e.  A  <->  ( jt  x )  e.  A
) )
1613, 15anbi12d 693 . . . . . . 7  |-  ( u  =  x  ->  (
( y  e.  u  /\  ( jt  u )  e.  A
)  <->  ( y  e.  x  /\  ( jt  x )  e.  A ) ) )
1716rspcev 3054 . . . . . 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 1183 . . . . 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 2800 . . . 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 17536 . . . 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 647 . . 3  |-  ( (
ph  /\  j  e.  A )  ->  j  e. Locally  A )
2221ex 425 . 2  |-  ( ph  ->  ( j  e.  A  ->  j  e. Locally  A )
)
2322ssrdv 3356 1  |-  ( ph  ->  A  C_ Locally  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    e. wcel 1726   A.wral 2707   E.wrex 2708    i^i cin 3321    C_ wss 3322   ~Pcpw 3801  (class class class)co 6084   ↾t crest 13653   Topctop 16963  Locally clly 17532
This theorem is referenced by:  llyidm  17556  nllyidm  17557  toplly  17558  hauslly  17560  lly1stc  17564
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-iota 5421  df-fv 5465  df-ov 6087  df-lly 17534
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