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Theorem opnregcld 25660
Description: A set is regularly closed iff it is the closure of some open set. (Contributed by Jeff Hankins, 27-Sep-2009.)
Hypothesis
Ref Expression
opnregcld.1  |-  X  = 
U. J
Assertion
Ref Expression
opnregcld  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( ( ( cls `  J ) `  (
( int `  J
) `  A )
)  =  A  <->  E. o  e.  J  A  =  ( ( cls `  J
) `  o )
) )
Distinct variable groups:    A, o    o, J    o, X

Proof of Theorem opnregcld
StepHypRef Expression
1 opnregcld.1 . . . . 5  |-  X  = 
U. J
21ntropn 16786 . . . 4  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( ( int `  J
) `  A )  e.  J )
3 eqcom 2285 . . . . 5  |-  ( ( ( cls `  J
) `  ( ( int `  J ) `  A ) )  =  A  <->  A  =  (
( cls `  J
) `  ( ( int `  J ) `  A ) ) )
43biimpi 186 . . . 4  |-  ( ( ( cls `  J
) `  ( ( int `  J ) `  A ) )  =  A  ->  A  =  ( ( cls `  J
) `  ( ( int `  J ) `  A ) ) )
5 fveq2 5525 . . . . . 6  |-  ( o  =  ( ( int `  J ) `  A
)  ->  ( ( cls `  J ) `  o )  =  ( ( cls `  J
) `  ( ( int `  J ) `  A ) ) )
65eqeq2d 2294 . . . . 5  |-  ( o  =  ( ( int `  J ) `  A
)  ->  ( A  =  ( ( cls `  J ) `  o
)  <->  A  =  (
( cls `  J
) `  ( ( int `  J ) `  A ) ) ) )
76rspcev 2884 . . . 4  |-  ( ( ( ( int `  J
) `  A )  e.  J  /\  A  =  ( ( cls `  J
) `  ( ( int `  J ) `  A ) ) )  ->  E. o  e.  J  A  =  ( ( cls `  J ) `  o ) )
82, 4, 7syl2an 463 . . 3  |-  ( ( ( J  e.  Top  /\  A  C_  X )  /\  ( ( cls `  J
) `  ( ( int `  J ) `  A ) )  =  A )  ->  E. o  e.  J  A  =  ( ( cls `  J
) `  o )
)
98ex 423 . 2  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( ( ( cls `  J ) `  (
( int `  J
) `  A )
)  =  A  ->  E. o  e.  J  A  =  ( ( cls `  J ) `  o ) ) )
10 simpl 443 . . . . . . . 8  |-  ( ( J  e.  Top  /\  o  e.  J )  ->  J  e.  Top )
111eltopss 16653 . . . . . . . . 9  |-  ( ( J  e.  Top  /\  o  e.  J )  ->  o  C_  X )
121clsss3 16796 . . . . . . . . 9  |-  ( ( J  e.  Top  /\  o  C_  X )  -> 
( ( cls `  J
) `  o )  C_  X )
1311, 12syldan 456 . . . . . . . 8  |-  ( ( J  e.  Top  /\  o  e.  J )  ->  ( ( cls `  J
) `  o )  C_  X )
141ntrss2 16794 . . . . . . . . 9  |-  ( ( J  e.  Top  /\  ( ( cls `  J
) `  o )  C_  X )  ->  (
( int `  J
) `  ( ( cls `  J ) `  o ) )  C_  ( ( cls `  J
) `  o )
)
1513, 14syldan 456 . . . . . . . 8  |-  ( ( J  e.  Top  /\  o  e.  J )  ->  ( ( int `  J
) `  ( ( cls `  J ) `  o ) )  C_  ( ( cls `  J
) `  o )
)
161clsss 16791 . . . . . . . 8  |-  ( ( J  e.  Top  /\  ( ( cls `  J
) `  o )  C_  X  /\  ( ( int `  J ) `
 ( ( cls `  J ) `  o
) )  C_  (
( cls `  J
) `  o )
)  ->  ( ( cls `  J ) `  ( ( int `  J
) `  ( ( cls `  J ) `  o ) ) ) 
C_  ( ( cls `  J ) `  (
( cls `  J
) `  o )
) )
1710, 13, 15, 16syl3anc 1182 . . . . . . 7  |-  ( ( J  e.  Top  /\  o  e.  J )  ->  ( ( cls `  J
) `  ( ( int `  J ) `  ( ( cls `  J
) `  o )
) )  C_  (
( cls `  J
) `  ( ( cls `  J ) `  o ) ) )
181clsidm 16804 . . . . . . . 8  |-  ( ( J  e.  Top  /\  o  C_  X )  -> 
( ( cls `  J
) `  ( ( cls `  J ) `  o ) )  =  ( ( cls `  J
) `  o )
)
1911, 18syldan 456 . . . . . . 7  |-  ( ( J  e.  Top  /\  o  e.  J )  ->  ( ( cls `  J
) `  ( ( cls `  J ) `  o ) )  =  ( ( cls `  J
) `  o )
)
2017, 19sseqtrd 3214 . . . . . 6  |-  ( ( J  e.  Top  /\  o  e.  J )  ->  ( ( cls `  J
) `  ( ( int `  J ) `  ( ( cls `  J
) `  o )
) )  C_  (
( cls `  J
) `  o )
)
211ntrss3 16797 . . . . . . . 8  |-  ( ( J  e.  Top  /\  ( ( cls `  J
) `  o )  C_  X )  ->  (
( int `  J
) `  ( ( cls `  J ) `  o ) )  C_  X )
2213, 21syldan 456 . . . . . . 7  |-  ( ( J  e.  Top  /\  o  e.  J )  ->  ( ( int `  J
) `  ( ( cls `  J ) `  o ) )  C_  X )
23 simpr 447 . . . . . . . 8  |-  ( ( J  e.  Top  /\  o  e.  J )  ->  o  e.  J )
241sscls 16793 . . . . . . . . 9  |-  ( ( J  e.  Top  /\  o  C_  X )  -> 
o  C_  ( ( cls `  J ) `  o ) )
2511, 24syldan 456 . . . . . . . 8  |-  ( ( J  e.  Top  /\  o  e.  J )  ->  o  C_  ( ( cls `  J ) `  o ) )
261ssntr 16795 . . . . . . . 8  |-  ( ( ( J  e.  Top  /\  ( ( cls `  J
) `  o )  C_  X )  /\  (
o  e.  J  /\  o  C_  ( ( cls `  J ) `  o
) ) )  -> 
o  C_  ( ( int `  J ) `  ( ( cls `  J
) `  o )
) )
2710, 13, 23, 25, 26syl22anc 1183 . . . . . . 7  |-  ( ( J  e.  Top  /\  o  e.  J )  ->  o  C_  ( ( int `  J ) `  ( ( cls `  J
) `  o )
) )
281clsss 16791 . . . . . . 7  |-  ( ( J  e.  Top  /\  ( ( int `  J
) `  ( ( cls `  J ) `  o ) )  C_  X  /\  o  C_  (
( int `  J
) `  ( ( cls `  J ) `  o ) ) )  ->  ( ( cls `  J ) `  o
)  C_  ( ( cls `  J ) `  ( ( int `  J
) `  ( ( cls `  J ) `  o ) ) ) )
2910, 22, 27, 28syl3anc 1182 . . . . . 6  |-  ( ( J  e.  Top  /\  o  e.  J )  ->  ( ( cls `  J
) `  o )  C_  ( ( cls `  J
) `  ( ( int `  J ) `  ( ( cls `  J
) `  o )
) ) )
3020, 29eqssd 3196 . . . . 5  |-  ( ( J  e.  Top  /\  o  e.  J )  ->  ( ( cls `  J
) `  ( ( int `  J ) `  ( ( cls `  J
) `  o )
) )  =  ( ( cls `  J
) `  o )
)
3130adantlr 695 . . . 4  |-  ( ( ( J  e.  Top  /\  A  C_  X )  /\  o  e.  J
)  ->  ( ( cls `  J ) `  ( ( int `  J
) `  ( ( cls `  J ) `  o ) ) )  =  ( ( cls `  J ) `  o
) )
32 fveq2 5525 . . . . . 6  |-  ( A  =  ( ( cls `  J ) `  o
)  ->  ( ( int `  J ) `  A )  =  ( ( int `  J
) `  ( ( cls `  J ) `  o ) ) )
3332fveq2d 5529 . . . . 5  |-  ( A  =  ( ( cls `  J ) `  o
)  ->  ( ( cls `  J ) `  ( ( int `  J
) `  A )
)  =  ( ( cls `  J ) `
 ( ( int `  J ) `  (
( cls `  J
) `  o )
) ) )
34 id 19 . . . . 5  |-  ( A  =  ( ( cls `  J ) `  o
)  ->  A  =  ( ( cls `  J
) `  o )
)
3533, 34eqeq12d 2297 . . . 4  |-  ( A  =  ( ( cls `  J ) `  o
)  ->  ( (
( cls `  J
) `  ( ( int `  J ) `  A ) )  =  A  <->  ( ( cls `  J ) `  (
( int `  J
) `  ( ( cls `  J ) `  o ) ) )  =  ( ( cls `  J ) `  o
) ) )
3631, 35syl5ibrcom 213 . . 3  |-  ( ( ( J  e.  Top  /\  A  C_  X )  /\  o  e.  J
)  ->  ( A  =  ( ( cls `  J ) `  o
)  ->  ( ( cls `  J ) `  ( ( int `  J
) `  A )
)  =  A ) )
3736rexlimdva 2667 . 2  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( E. o  e.  J  A  =  ( ( cls `  J
) `  o )  ->  ( ( cls `  J
) `  ( ( int `  J ) `  A ) )  =  A ) )
389, 37impbid 183 1  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( ( ( cls `  J ) `  (
( int `  J
) `  A )
)  =  A  <->  E. o  e.  J  A  =  ( ( cls `  J
) `  o )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   E.wrex 2544    C_ wss 3152   U.cuni 3827   ` cfv 5255   Topctop 16631   intcnt 16754   clsccl 16755
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-top 16636  df-cld 16756  df-ntr 16757  df-cls 16758
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