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Theorem toponmre 17162
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 17065. (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 16997 . . . . . 6  |-  ( b  e.  (TopOn `  B
)  ->  B  =  U. b )
2 eqimss2 3403 . . . . . . 7  |-  ( B  =  U. b  ->  U. b  C_  B )
3 sspwuni 4179 . . . . . . 7  |-  ( b 
C_  ~P B  <->  U. b  C_  B )
42, 3sylibr 205 . . . . . 6  |-  ( B  =  U. b  -> 
b  C_  ~P B
)
51, 4syl 16 . . . . 5  |-  ( b  e.  (TopOn `  B
)  ->  b  C_  ~P B )
6 vex 2961 . . . . . 6  |-  b  e. 
_V
76elpw 3807 . . . . 5  |-  ( b  e.  ~P ~P B  <->  b 
C_  ~P B )
85, 7sylibr 205 . . . 4  |-  ( b  e.  (TopOn `  B
)  ->  b  e.  ~P ~P B )
98ssriv 3354 . . 3  |-  (TopOn `  B )  C_  ~P ~P B
109a1i 11 . 2  |-  ( B  e.  V  ->  (TopOn `  B )  C_  ~P ~P B )
11 distopon 17066 . 2  |-  ( B  e.  V  ->  ~P B  e.  (TopOn `  B
) )
12 simpl 445 . . . . . . . . . . . . . 14  |-  ( ( b  C_  (TopOn `  B
)  /\  b  =/=  (/) )  ->  b  C_  (TopOn `  B ) )
1312sselda 3350 . . . . . . . . . . . . 13  |-  ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  x  e.  b )  ->  x  e.  (TopOn `  B )
)
1413adantrl 698 . . . . . . . . . . . 12  |-  ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  (
c  C_  |^| b  /\  x  e.  b )
)  ->  x  e.  (TopOn `  B ) )
15 topontop 16996 . . . . . . . . . . . 12  |-  ( x  e.  (TopOn `  B
)  ->  x  e.  Top )
1614, 15syl 16 . . . . . . . . . . 11  |-  ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  (
c  C_  |^| b  /\  x  e.  b )
)  ->  x  e.  Top )
17 simpl 445 . . . . . . . . . . . . 13  |-  ( ( c  C_  |^| b  /\  x  e.  b )  ->  c  C_  |^| b )
18 intss1 4067 . . . . . . . . . . . . . 14  |-  ( x  e.  b  ->  |^| b  C_  x )
1918adantl 454 . . . . . . . . . . . . 13  |-  ( ( c  C_  |^| b  /\  x  e.  b )  ->  |^| b  C_  x
)
2017, 19sstrd 3360 . . . . . . . . . . . 12  |-  ( ( c  C_  |^| b  /\  x  e.  b )  ->  c  C_  x )
2120adantl 454 . . . . . . . . . . 11  |-  ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  (
c  C_  |^| b  /\  x  e.  b )
)  ->  c  C_  x )
22 uniopn 16975 . . . . . . . . . . 11  |-  ( ( x  e.  Top  /\  c  C_  x )  ->  U. c  e.  x
)
2316, 21, 22syl2anc 644 . . . . . . . . . 10  |-  ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  (
c  C_  |^| b  /\  x  e.  b )
)  ->  U. c  e.  x )
2423expr 600 . . . . . . . . 9  |-  ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  c  C_ 
|^| b )  -> 
( x  e.  b  ->  U. c  e.  x
) )
2524ralrimiv 2790 . . . . . . . 8  |-  ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  c  C_ 
|^| b )  ->  A. x  e.  b  U. c  e.  x
)
26 vex 2961 . . . . . . . . . 10  |-  c  e. 
_V
2726uniex 4708 . . . . . . . . 9  |-  U. c  e.  _V
2827elint2 4059 . . . . . . . 8  |-  ( U. c  e.  |^| b  <->  A. x  e.  b  U. c  e.  x )
2925, 28sylibr 205 . . . . . . 7  |-  ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  c  C_ 
|^| b )  ->  U. c  e.  |^| b
)
3029ex 425 . . . . . 6  |-  ( ( b  C_  (TopOn `  B
)  /\  b  =/=  (/) )  ->  ( c  C_ 
|^| b  ->  U. c  e.  |^| b ) )
3130alrimiv 1642 . . . . 5  |-  ( ( b  C_  (TopOn `  B
)  /\  b  =/=  (/) )  ->  A. c
( c  C_  |^| b  ->  U. c  e.  |^| b ) )
32 simpll 732 . . . . . . . . . . 11  |-  ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  (
c  e.  |^| b  /\  x  e.  |^| b
) )  ->  b  C_  (TopOn `  B )
)
3332sselda 3350 . . . . . . . . . 10  |-  ( ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  (
c  e.  |^| b  /\  x  e.  |^| b
) )  /\  y  e.  b )  ->  y  e.  (TopOn `  B )
)
34 topontop 16996 . . . . . . . . . 10  |-  ( y  e.  (TopOn `  B
)  ->  y  e.  Top )
3533, 34syl 16 . . . . . . . . 9  |-  ( ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  (
c  e.  |^| b  /\  x  e.  |^| b
) )  /\  y  e.  b )  ->  y  e.  Top )
36 intss1 4067 . . . . . . . . . . 11  |-  ( y  e.  b  ->  |^| b  C_  y )
3736adantl 454 . . . . . . . . . 10  |-  ( ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  (
c  e.  |^| b  /\  x  e.  |^| b
) )  /\  y  e.  b )  ->  |^| b  C_  y )
38 simplrl 738 . . . . . . . . . 10  |-  ( ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  (
c  e.  |^| b  /\  x  e.  |^| b
) )  /\  y  e.  b )  ->  c  e.  |^| b )
3937, 38sseldd 3351 . . . . . . . . 9  |-  ( ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  (
c  e.  |^| b  /\  x  e.  |^| b
) )  /\  y  e.  b )  ->  c  e.  y )
40 simplrr 739 . . . . . . . . . 10  |-  ( ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  (
c  e.  |^| b  /\  x  e.  |^| b
) )  /\  y  e.  b )  ->  x  e.  |^| b )
4137, 40sseldd 3351 . . . . . . . . 9  |-  ( ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  (
c  e.  |^| b  /\  x  e.  |^| b
) )  /\  y  e.  b )  ->  x  e.  y )
42 inopn 16977 . . . . . . . . 9  |-  ( ( y  e.  Top  /\  c  e.  y  /\  x  e.  y )  ->  ( c  i^i  x
)  e.  y )
4335, 39, 41, 42syl3anc 1185 . . . . . . . 8  |-  ( ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  (
c  e.  |^| b  /\  x  e.  |^| b
) )  /\  y  e.  b )  ->  (
c  i^i  x )  e.  y )
4443ralrimiva 2791 . . . . . . 7  |-  ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  (
c  e.  |^| b  /\  x  e.  |^| b
) )  ->  A. y  e.  b  ( c  i^i  x )  e.  y )
4526inex1 4347 . . . . . . . 8  |-  ( c  i^i  x )  e. 
_V
4645elint2 4059 . . . . . . 7  |-  ( ( c  i^i  x )  e.  |^| b  <->  A. y  e.  b  ( c  i^i  x )  e.  y )
4744, 46sylibr 205 . . . . . 6  |-  ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  (
c  e.  |^| b  /\  x  e.  |^| b
) )  ->  (
c  i^i  x )  e.  |^| b )
4847ralrimivva 2800 . . . . 5  |-  ( ( b  C_  (TopOn `  B
)  /\  b  =/=  (/) )  ->  A. c  e.  |^| b A. x  e.  |^| b ( c  i^i  x )  e. 
|^| b )
49 intex 4359 . . . . . . . 8  |-  ( b  =/=  (/)  <->  |^| b  e.  _V )
5049biimpi 188 . . . . . . 7  |-  ( b  =/=  (/)  ->  |^| b  e. 
_V )
5150adantl 454 . . . . . 6  |-  ( ( b  C_  (TopOn `  B
)  /\  b  =/=  (/) )  ->  |^| b  e. 
_V )
52 istopg 16973 . . . . . 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 16 . . . . 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 890 . . . 4  |-  ( ( b  C_  (TopOn `  B
)  /\  b  =/=  (/) )  ->  |^| b  e. 
Top )
55543adant1 976 . . 3  |-  ( ( B  e.  V  /\  b  C_  (TopOn `  B
)  /\  b  =/=  (/) )  ->  |^| b  e. 
Top )
56 n0 3639 . . . . . . . . . . 11  |-  ( b  =/=  (/)  <->  E. x  x  e.  b )
5756biimpi 188 . . . . . . . . . 10  |-  ( b  =/=  (/)  ->  E. x  x  e.  b )
5857ad2antlr 709 . . . . . . . . 9  |-  ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  c  e.  |^| b )  ->  E. x  x  e.  b )
5918sselda 3350 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  b  /\  c  e.  |^| b )  ->  c  e.  x
)
6059ancoms 441 . . . . . . . . . . . . . 14  |-  ( ( c  e.  |^| b  /\  x  e.  b
)  ->  c  e.  x )
61 elssuni 4045 . . . . . . . . . . . . . 14  |-  ( c  e.  x  ->  c  C_ 
U. x )
6260, 61syl 16 . . . . . . . . . . . . 13  |-  ( ( c  e.  |^| b  /\  x  e.  b
)  ->  c  C_  U. x )
6362adantl 454 . . . . . . . . . . . 12  |-  ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  (
c  e.  |^| b  /\  x  e.  b
) )  ->  c  C_ 
U. x )
6413adantrl 698 . . . . . . . . . . . . 13  |-  ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  (
c  e.  |^| b  /\  x  e.  b
) )  ->  x  e.  (TopOn `  B )
)
65 toponuni 16997 . . . . . . . . . . . . 13  |-  ( x  e.  (TopOn `  B
)  ->  B  =  U. x )
6664, 65syl 16 . . . . . . . . . . . 12  |-  ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  (
c  e.  |^| b  /\  x  e.  b
) )  ->  B  =  U. x )
6763, 66sseqtr4d 3387 . . . . . . . . . . 11  |-  ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  (
c  e.  |^| b  /\  x  e.  b
) )  ->  c  C_  B )
6867expr 600 . . . . . . . . . 10  |-  ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  c  e.  |^| b )  -> 
( x  e.  b  ->  c  C_  B
) )
6968exlimdv 1647 . . . . . . . . 9  |-  ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  c  e.  |^| b )  -> 
( E. x  x  e.  b  ->  c  C_  B ) )
7058, 69mpd 15 . . . . . . . 8  |-  ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  c  e.  |^| b )  -> 
c  C_  B )
7170ralrimiva 2791 . . . . . . 7  |-  ( ( b  C_  (TopOn `  B
)  /\  b  =/=  (/) )  ->  A. c  e.  |^| b c  C_  B )
72 unissb 4047 . . . . . . 7  |-  ( U. |^| b  C_  B  <->  A. c  e.  |^| b c  C_  B )
7371, 72sylibr 205 . . . . . 6  |-  ( ( b  C_  (TopOn `  B
)  /\  b  =/=  (/) )  ->  U. |^| b  C_  B )
74733adant1 976 . . . . 5  |-  ( ( B  e.  V  /\  b  C_  (TopOn `  B
)  /\  b  =/=  (/) )  ->  U. |^| b  C_  B )
7512sselda 3350 . . . . . . . . . 10  |-  ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  c  e.  b )  ->  c  e.  (TopOn `  B )
)
76 toponuni 16997 . . . . . . . . . 10  |-  ( c  e.  (TopOn `  B
)  ->  B  =  U. c )
7775, 76syl 16 . . . . . . . . 9  |-  ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  c  e.  b )  ->  B  =  U. c )
78 topontop 16996 . . . . . . . . . 10  |-  ( c  e.  (TopOn `  B
)  ->  c  e.  Top )
79 eqid 2438 . . . . . . . . . . 11  |-  U. c  =  U. c
8079topopn 16984 . . . . . . . . . 10  |-  ( c  e.  Top  ->  U. c  e.  c )
8175, 78, 803syl 19 . . . . . . . . 9  |-  ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  c  e.  b )  ->  U. c  e.  c )
8277, 81eqeltrd 2512 . . . . . . . 8  |-  ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  c  e.  b )  ->  B  e.  c )
8382ralrimiva 2791 . . . . . . 7  |-  ( ( b  C_  (TopOn `  B
)  /\  b  =/=  (/) )  ->  A. c  e.  b  B  e.  c )
84833adant1 976 . . . . . 6  |-  ( ( B  e.  V  /\  b  C_  (TopOn `  B
)  /\  b  =/=  (/) )  ->  A. c  e.  b  B  e.  c )
85 elintg 4060 . . . . . . 7  |-  ( B  e.  V  ->  ( B  e.  |^| b  <->  A. c  e.  b  B  e.  c ) )
86853ad2ant1 979 . . . . . 6  |-  ( ( B  e.  V  /\  b  C_  (TopOn `  B
)  /\  b  =/=  (/) )  ->  ( B  e.  |^| b  <->  A. c  e.  b  B  e.  c ) )
8784, 86mpbird 225 . . . . 5  |-  ( ( B  e.  V  /\  b  C_  (TopOn `  B
)  /\  b  =/=  (/) )  ->  B  e.  |^| b )
88 unissel 4046 . . . . 5  |-  ( ( U. |^| b  C_  B  /\  B  e.  |^| b )  ->  U. |^| b  =  B )
8974, 87, 88syl2anc 644 . . . 4  |-  ( ( B  e.  V  /\  b  C_  (TopOn `  B
)  /\  b  =/=  (/) )  ->  U. |^| b  =  B )
9089eqcomd 2443 . . 3  |-  ( ( B  e.  V  /\  b  C_  (TopOn `  B
)  /\  b  =/=  (/) )  ->  B  =  U. |^| b )
91 istopon 16995 . . 3  |-  ( |^| b  e.  (TopOn `  B
)  <->  ( |^| b  e.  Top  /\  B  = 
U. |^| b ) )
9255, 90, 91sylanbrc 647 . 2  |-  ( ( B  e.  V  /\  b  C_  (TopOn `  B
)  /\  b  =/=  (/) )  ->  |^| b  e.  (TopOn `  B )
)
9310, 11, 92ismred 13832 1  |-  ( B  e.  V  ->  (TopOn `  B )  e.  (Moore `  ~P B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937   A.wal 1550   E.wex 1551    = wceq 1653    e. wcel 1726    =/= wne 2601   A.wral 2707   _Vcvv 2958    i^i cin 3321    C_ wss 3322   (/)c0 3630   ~Pcpw 3801   U.cuni 4017   |^|cint 4052   ` cfv 5457  Moorecmre 13812   Topctop 16963  TopOnctopon 16964
This theorem is referenced by:  topmtcl  26406
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-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
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-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  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-int 4053  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-iota 5421  df-fun 5459  df-fv 5465  df-mre 13816  df-top 16968  df-topon 16971
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