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Theorem toponmre 16830
Description: The topologies over a given base set form a Moore collection: the intersection of any family of them is a topology, including the empty (relative) intersection which gives the discrete topology distop 16733. (Contributed by Stefan O'Rear, 31-Jan-2015.) (Revised by Mario Carneiro, 5-May-2015.)
Assertion
Ref Expression
toponmre  |-  ( B  e.  V  ->  (TopOn `  B )  e.  (Moore `  ~P B ) )

Proof of Theorem toponmre
Dummy variables  b 
c  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 toponuni 16665 . . . . . 6  |-  ( b  e.  (TopOn `  B
)  ->  B  =  U. b )
2 eqimss2 3231 . . . . . . 7  |-  ( B  =  U. b  ->  U. b  C_  B )
3 sspwuni 3987 . . . . . . 7  |-  ( b 
C_  ~P B  <->  U. b  C_  B )
42, 3sylibr 203 . . . . . 6  |-  ( B  =  U. b  -> 
b  C_  ~P B
)
51, 4syl 15 . . . . 5  |-  ( b  e.  (TopOn `  B
)  ->  b  C_  ~P B )
6 vex 2791 . . . . . 6  |-  b  e. 
_V
76elpw 3631 . . . . 5  |-  ( b  e.  ~P ~P B  <->  b 
C_  ~P B )
85, 7sylibr 203 . . . 4  |-  ( b  e.  (TopOn `  B
)  ->  b  e.  ~P ~P B )
98ssriv 3184 . . 3  |-  (TopOn `  B )  C_  ~P ~P B
109a1i 10 . 2  |-  ( B  e.  V  ->  (TopOn `  B )  C_  ~P ~P B )
11 distopon 16734 . 2  |-  ( B  e.  V  ->  ~P B  e.  (TopOn `  B
) )
12 simpl 443 . . . . . . . . . . . . . 14  |-  ( ( b  C_  (TopOn `  B
)  /\  b  =/=  (/) )  ->  b  C_  (TopOn `  B ) )
1312sselda 3180 . . . . . . . . . . . . 13  |-  ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  x  e.  b )  ->  x  e.  (TopOn `  B )
)
1413adantrl 696 . . . . . . . . . . . 12  |-  ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  (
c  C_  |^| b  /\  x  e.  b )
)  ->  x  e.  (TopOn `  B ) )
15 topontop 16664 . . . . . . . . . . . 12  |-  ( x  e.  (TopOn `  B
)  ->  x  e.  Top )
1614, 15syl 15 . . . . . . . . . . 11  |-  ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  (
c  C_  |^| b  /\  x  e.  b )
)  ->  x  e.  Top )
17 simpl 443 . . . . . . . . . . . . 13  |-  ( ( c  C_  |^| b  /\  x  e.  b )  ->  c  C_  |^| b )
18 intss1 3877 . . . . . . . . . . . . . 14  |-  ( x  e.  b  ->  |^| b  C_  x )
1918adantl 452 . . . . . . . . . . . . 13  |-  ( ( c  C_  |^| b  /\  x  e.  b )  ->  |^| b  C_  x
)
2017, 19sstrd 3189 . . . . . . . . . . . 12  |-  ( ( c  C_  |^| b  /\  x  e.  b )  ->  c  C_  x )
2120adantl 452 . . . . . . . . . . 11  |-  ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  (
c  C_  |^| b  /\  x  e.  b )
)  ->  c  C_  x )
22 uniopn 16643 . . . . . . . . . . 11  |-  ( ( x  e.  Top  /\  c  C_  x )  ->  U. c  e.  x
)
2316, 21, 22syl2anc 642 . . . . . . . . . 10  |-  ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  (
c  C_  |^| b  /\  x  e.  b )
)  ->  U. c  e.  x )
2423expr 598 . . . . . . . . 9  |-  ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  c  C_ 
|^| b )  -> 
( x  e.  b  ->  U. c  e.  x
) )
2524ralrimiv 2625 . . . . . . . 8  |-  ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  c  C_ 
|^| b )  ->  A. x  e.  b  U. c  e.  x
)
26 vex 2791 . . . . . . . . . 10  |-  c  e. 
_V
2726uniex 4516 . . . . . . . . 9  |-  U. c  e.  _V
2827elint2 3869 . . . . . . . 8  |-  ( U. c  e.  |^| b  <->  A. x  e.  b  U. c  e.  x )
2925, 28sylibr 203 . . . . . . 7  |-  ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  c  C_ 
|^| b )  ->  U. c  e.  |^| b
)
3029ex 423 . . . . . 6  |-  ( ( b  C_  (TopOn `  B
)  /\  b  =/=  (/) )  ->  ( c  C_ 
|^| b  ->  U. c  e.  |^| b ) )
3130alrimiv 1617 . . . . 5  |-  ( ( b  C_  (TopOn `  B
)  /\  b  =/=  (/) )  ->  A. c
( c  C_  |^| b  ->  U. c  e.  |^| b ) )
32 simpll 730 . . . . . . . . . . 11  |-  ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  (
c  e.  |^| b  /\  x  e.  |^| b
) )  ->  b  C_  (TopOn `  B )
)
3332sselda 3180 . . . . . . . . . 10  |-  ( ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  (
c  e.  |^| b  /\  x  e.  |^| b
) )  /\  y  e.  b )  ->  y  e.  (TopOn `  B )
)
34 topontop 16664 . . . . . . . . . 10  |-  ( y  e.  (TopOn `  B
)  ->  y  e.  Top )
3533, 34syl 15 . . . . . . . . 9  |-  ( ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  (
c  e.  |^| b  /\  x  e.  |^| b
) )  /\  y  e.  b )  ->  y  e.  Top )
36 intss1 3877 . . . . . . . . . . 11  |-  ( y  e.  b  ->  |^| b  C_  y )
3736adantl 452 . . . . . . . . . 10  |-  ( ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  (
c  e.  |^| b  /\  x  e.  |^| b
) )  /\  y  e.  b )  ->  |^| b  C_  y )
38 simplrl 736 . . . . . . . . . 10  |-  ( ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  (
c  e.  |^| b  /\  x  e.  |^| b
) )  /\  y  e.  b )  ->  c  e.  |^| b )
3937, 38sseldd 3181 . . . . . . . . 9  |-  ( ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  (
c  e.  |^| b  /\  x  e.  |^| b
) )  /\  y  e.  b )  ->  c  e.  y )
40 simplrr 737 . . . . . . . . . 10  |-  ( ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  (
c  e.  |^| b  /\  x  e.  |^| b
) )  /\  y  e.  b )  ->  x  e.  |^| b )
4137, 40sseldd 3181 . . . . . . . . 9  |-  ( ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  (
c  e.  |^| b  /\  x  e.  |^| b
) )  /\  y  e.  b )  ->  x  e.  y )
42 inopn 16645 . . . . . . . . 9  |-  ( ( y  e.  Top  /\  c  e.  y  /\  x  e.  y )  ->  ( c  i^i  x
)  e.  y )
4335, 39, 41, 42syl3anc 1182 . . . . . . . 8  |-  ( ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  (
c  e.  |^| b  /\  x  e.  |^| b
) )  /\  y  e.  b )  ->  (
c  i^i  x )  e.  y )
4443ralrimiva 2626 . . . . . . 7  |-  ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  (
c  e.  |^| b  /\  x  e.  |^| b
) )  ->  A. y  e.  b  ( c  i^i  x )  e.  y )
4526inex1 4155 . . . . . . . 8  |-  ( c  i^i  x )  e. 
_V
4645elint2 3869 . . . . . . 7  |-  ( ( c  i^i  x )  e.  |^| b  <->  A. y  e.  b  ( c  i^i  x )  e.  y )
4744, 46sylibr 203 . . . . . 6  |-  ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  (
c  e.  |^| b  /\  x  e.  |^| b
) )  ->  (
c  i^i  x )  e.  |^| b )
4847ralrimivva 2635 . . . . 5  |-  ( ( b  C_  (TopOn `  B
)  /\  b  =/=  (/) )  ->  A. c  e.  |^| b A. x  e.  |^| b ( c  i^i  x )  e. 
|^| b )
49 intex 4167 . . . . . . . 8  |-  ( b  =/=  (/)  <->  |^| b  e.  _V )
5049biimpi 186 . . . . . . 7  |-  ( b  =/=  (/)  ->  |^| b  e. 
_V )
5150adantl 452 . . . . . 6  |-  ( ( b  C_  (TopOn `  B
)  /\  b  =/=  (/) )  ->  |^| b  e. 
_V )
52 istopg 16641 . . . . . 6  |-  ( |^| b  e.  _V  ->  (
|^| b  e.  Top  <->  ( A. c ( c  C_  |^| b  ->  U. c  e.  |^| b )  /\  A. c  e.  |^| b A. x  e.  |^| b
( c  i^i  x
)  e.  |^| b
) ) )
5351, 52syl 15 . . . . 5  |-  ( ( b  C_  (TopOn `  B
)  /\  b  =/=  (/) )  ->  ( |^| b  e.  Top  <->  ( A. c ( c  C_  |^| b  ->  U. c  e.  |^| b )  /\  A. c  e.  |^| b A. x  e.  |^| b
( c  i^i  x
)  e.  |^| b
) ) )
5431, 48, 53mpbir2and 888 . . . 4  |-  ( ( b  C_  (TopOn `  B
)  /\  b  =/=  (/) )  ->  |^| b  e. 
Top )
55543adant1 973 . . 3  |-  ( ( B  e.  V  /\  b  C_  (TopOn `  B
)  /\  b  =/=  (/) )  ->  |^| b  e. 
Top )
56 n0 3464 . . . . . . . . . . 11  |-  ( b  =/=  (/)  <->  E. x  x  e.  b )
5756biimpi 186 . . . . . . . . . 10  |-  ( b  =/=  (/)  ->  E. x  x  e.  b )
5857ad2antlr 707 . . . . . . . . 9  |-  ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  c  e.  |^| b )  ->  E. x  x  e.  b )
5918sselda 3180 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  b  /\  c  e.  |^| b )  ->  c  e.  x
)
6059ancoms 439 . . . . . . . . . . . . . 14  |-  ( ( c  e.  |^| b  /\  x  e.  b
)  ->  c  e.  x )
61 elssuni 3855 . . . . . . . . . . . . . 14  |-  ( c  e.  x  ->  c  C_ 
U. x )
6260, 61syl 15 . . . . . . . . . . . . 13  |-  ( ( c  e.  |^| b  /\  x  e.  b
)  ->  c  C_  U. x )
6362adantl 452 . . . . . . . . . . . 12  |-  ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  (
c  e.  |^| b  /\  x  e.  b
) )  ->  c  C_ 
U. x )
6413adantrl 696 . . . . . . . . . . . . 13  |-  ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  (
c  e.  |^| b  /\  x  e.  b
) )  ->  x  e.  (TopOn `  B )
)
65 toponuni 16665 . . . . . . . . . . . . 13  |-  ( x  e.  (TopOn `  B
)  ->  B  =  U. x )
6664, 65syl 15 . . . . . . . . . . . 12  |-  ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  (
c  e.  |^| b  /\  x  e.  b
) )  ->  B  =  U. x )
6763, 66sseqtr4d 3215 . . . . . . . . . . 11  |-  ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  (
c  e.  |^| b  /\  x  e.  b
) )  ->  c  C_  B )
6867expr 598 . . . . . . . . . 10  |-  ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  c  e.  |^| b )  -> 
( x  e.  b  ->  c  C_  B
) )
6968exlimdv 1664 . . . . . . . . 9  |-  ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  c  e.  |^| b )  -> 
( E. x  x  e.  b  ->  c  C_  B ) )
7058, 69mpd 14 . . . . . . . 8  |-  ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  c  e.  |^| b )  -> 
c  C_  B )
7170ralrimiva 2626 . . . . . . 7  |-  ( ( b  C_  (TopOn `  B
)  /\  b  =/=  (/) )  ->  A. c  e.  |^| b c  C_  B )
72 unissb 3857 . . . . . . 7  |-  ( U. |^| b  C_  B  <->  A. c  e.  |^| b c  C_  B )
7371, 72sylibr 203 . . . . . 6  |-  ( ( b  C_  (TopOn `  B
)  /\  b  =/=  (/) )  ->  U. |^| b  C_  B )
74733adant1 973 . . . . 5  |-  ( ( B  e.  V  /\  b  C_  (TopOn `  B
)  /\  b  =/=  (/) )  ->  U. |^| b  C_  B )
7512sselda 3180 . . . . . . . . . 10  |-  ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  c  e.  b )  ->  c  e.  (TopOn `  B )
)
76 toponuni 16665 . . . . . . . . . 10  |-  ( c  e.  (TopOn `  B
)  ->  B  =  U. c )
7775, 76syl 15 . . . . . . . . 9  |-  ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  c  e.  b )  ->  B  =  U. c )
78 topontop 16664 . . . . . . . . . 10  |-  ( c  e.  (TopOn `  B
)  ->  c  e.  Top )
79 eqid 2283 . . . . . . . . . . 11  |-  U. c  =  U. c
8079topopn 16652 . . . . . . . . . 10  |-  ( c  e.  Top  ->  U. c  e.  c )
8175, 78, 803syl 18 . . . . . . . . 9  |-  ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  c  e.  b )  ->  U. c  e.  c )
8277, 81eqeltrd 2357 . . . . . . . 8  |-  ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  c  e.  b )  ->  B  e.  c )
8382ralrimiva 2626 . . . . . . 7  |-  ( ( b  C_  (TopOn `  B
)  /\  b  =/=  (/) )  ->  A. c  e.  b  B  e.  c )
84833adant1 973 . . . . . 6  |-  ( ( B  e.  V  /\  b  C_  (TopOn `  B
)  /\  b  =/=  (/) )  ->  A. c  e.  b  B  e.  c )
85 elintg 3870 . . . . . . 7  |-  ( B  e.  V  ->  ( B  e.  |^| b  <->  A. c  e.  b  B  e.  c ) )
86853ad2ant1 976 . . . . . 6  |-  ( ( B  e.  V  /\  b  C_  (TopOn `  B
)  /\  b  =/=  (/) )  ->  ( B  e.  |^| b  <->  A. c  e.  b  B  e.  c ) )
8784, 86mpbird 223 . . . . 5  |-  ( ( B  e.  V  /\  b  C_  (TopOn `  B
)  /\  b  =/=  (/) )  ->  B  e.  |^| b )
88 unissel 3856 . . . . 5  |-  ( ( U. |^| b  C_  B  /\  B  e.  |^| b )  ->  U. |^| b  =  B )
8974, 87, 88syl2anc 642 . . . 4  |-  ( ( B  e.  V  /\  b  C_  (TopOn `  B
)  /\  b  =/=  (/) )  ->  U. |^| b  =  B )
9089eqcomd 2288 . . 3  |-  ( ( B  e.  V  /\  b  C_  (TopOn `  B
)  /\  b  =/=  (/) )  ->  B  =  U. |^| b )
91 istopon 16663 . . 3  |-  ( |^| b  e.  (TopOn `  B
)  <->  ( |^| b  e.  Top  /\  B  = 
U. |^| b ) )
9255, 90, 91sylanbrc 645 . 2  |-  ( ( B  e.  V  /\  b  C_  (TopOn `  B
)  /\  b  =/=  (/) )  ->  |^| b  e.  (TopOn `  B )
)
9310, 11, 92ismred 13504 1  |-  ( B  e.  V  ->  (TopOn `  B )  e.  (Moore `  ~P B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934   A.wal 1527   E.wex 1528    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   _Vcvv 2788    i^i cin 3151    C_ wss 3152   (/)c0 3455   ~Pcpw 3625   U.cuni 3827   |^|cint 3862   ` cfv 5255  Moorecmre 13484   Topctop 16631  TopOnctopon 16632
This theorem is referenced by:  topmtcl  26312
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-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-rab 2552  df-v 2790  df-sbc 2992  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-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-iota 5219  df-fun 5257  df-fv 5263  df-mre 13488  df-top 16636  df-topon 16639
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