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Theorem toponmre 17120
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 17023. (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 16955 . . . . . 6  |-  ( b  e.  (TopOn `  B
)  ->  B  =  U. b )
2 eqimss2 3369 . . . . . . 7  |-  ( B  =  U. b  ->  U. b  C_  B )
3 sspwuni 4144 . . . . . . 7  |-  ( b 
C_  ~P B  <->  U. b  C_  B )
42, 3sylibr 204 . . . . . 6  |-  ( B  =  U. b  -> 
b  C_  ~P B
)
51, 4syl 16 . . . . 5  |-  ( b  e.  (TopOn `  B
)  ->  b  C_  ~P B )
6 vex 2927 . . . . . 6  |-  b  e. 
_V
76elpw 3773 . . . . 5  |-  ( b  e.  ~P ~P B  <->  b 
C_  ~P B )
85, 7sylibr 204 . . . 4  |-  ( b  e.  (TopOn `  B
)  ->  b  e.  ~P ~P B )
98ssriv 3320 . . 3  |-  (TopOn `  B )  C_  ~P ~P B
109a1i 11 . 2  |-  ( B  e.  V  ->  (TopOn `  B )  C_  ~P ~P B )
11 distopon 17024 . 2  |-  ( B  e.  V  ->  ~P B  e.  (TopOn `  B
) )
12 simpl 444 . . . . . . . . . . . . . 14  |-  ( ( b  C_  (TopOn `  B
)  /\  b  =/=  (/) )  ->  b  C_  (TopOn `  B ) )
1312sselda 3316 . . . . . . . . . . . . 13  |-  ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  x  e.  b )  ->  x  e.  (TopOn `  B )
)
1413adantrl 697 . . . . . . . . . . . 12  |-  ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  (
c  C_  |^| b  /\  x  e.  b )
)  ->  x  e.  (TopOn `  B ) )
15 topontop 16954 . . . . . . . . . . . 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 444 . . . . . . . . . . . . 13  |-  ( ( c  C_  |^| b  /\  x  e.  b )  ->  c  C_  |^| b )
18 intss1 4033 . . . . . . . . . . . . . 14  |-  ( x  e.  b  ->  |^| b  C_  x )
1918adantl 453 . . . . . . . . . . . . 13  |-  ( ( c  C_  |^| b  /\  x  e.  b )  ->  |^| b  C_  x
)
2017, 19sstrd 3326 . . . . . . . . . . . 12  |-  ( ( c  C_  |^| b  /\  x  e.  b )  ->  c  C_  x )
2120adantl 453 . . . . . . . . . . 11  |-  ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  (
c  C_  |^| b  /\  x  e.  b )
)  ->  c  C_  x )
22 uniopn 16933 . . . . . . . . . . 11  |-  ( ( x  e.  Top  /\  c  C_  x )  ->  U. c  e.  x
)
2316, 21, 22syl2anc 643 . . . . . . . . . 10  |-  ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  (
c  C_  |^| b  /\  x  e.  b )
)  ->  U. c  e.  x )
2423expr 599 . . . . . . . . 9  |-  ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  c  C_ 
|^| b )  -> 
( x  e.  b  ->  U. c  e.  x
) )
2524ralrimiv 2756 . . . . . . . 8  |-  ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  c  C_ 
|^| b )  ->  A. x  e.  b  U. c  e.  x
)
26 vex 2927 . . . . . . . . . 10  |-  c  e. 
_V
2726uniex 4672 . . . . . . . . 9  |-  U. c  e.  _V
2827elint2 4025 . . . . . . . 8  |-  ( U. c  e.  |^| b  <->  A. x  e.  b  U. c  e.  x )
2925, 28sylibr 204 . . . . . . 7  |-  ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  c  C_ 
|^| b )  ->  U. c  e.  |^| b
)
3029ex 424 . . . . . 6  |-  ( ( b  C_  (TopOn `  B
)  /\  b  =/=  (/) )  ->  ( c  C_ 
|^| b  ->  U. c  e.  |^| b ) )
3130alrimiv 1638 . . . . 5  |-  ( ( b  C_  (TopOn `  B
)  /\  b  =/=  (/) )  ->  A. c
( c  C_  |^| b  ->  U. c  e.  |^| b ) )
32 simpll 731 . . . . . . . . . . 11  |-  ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  (
c  e.  |^| b  /\  x  e.  |^| b
) )  ->  b  C_  (TopOn `  B )
)
3332sselda 3316 . . . . . . . . . 10  |-  ( ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  (
c  e.  |^| b  /\  x  e.  |^| b
) )  /\  y  e.  b )  ->  y  e.  (TopOn `  B )
)
34 topontop 16954 . . . . . . . . . 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 4033 . . . . . . . . . . 11  |-  ( y  e.  b  ->  |^| b  C_  y )
3736adantl 453 . . . . . . . . . 10  |-  ( ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  (
c  e.  |^| b  /\  x  e.  |^| b
) )  /\  y  e.  b )  ->  |^| b  C_  y )
38 simplrl 737 . . . . . . . . . 10  |-  ( ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  (
c  e.  |^| b  /\  x  e.  |^| b
) )  /\  y  e.  b )  ->  c  e.  |^| b )
3937, 38sseldd 3317 . . . . . . . . 9  |-  ( ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  (
c  e.  |^| b  /\  x  e.  |^| b
) )  /\  y  e.  b )  ->  c  e.  y )
40 simplrr 738 . . . . . . . . . 10  |-  ( ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  (
c  e.  |^| b  /\  x  e.  |^| b
) )  /\  y  e.  b )  ->  x  e.  |^| b )
4137, 40sseldd 3317 . . . . . . . . 9  |-  ( ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  (
c  e.  |^| b  /\  x  e.  |^| b
) )  /\  y  e.  b )  ->  x  e.  y )
42 inopn 16935 . . . . . . . . 9  |-  ( ( y  e.  Top  /\  c  e.  y  /\  x  e.  y )  ->  ( c  i^i  x
)  e.  y )
4335, 39, 41, 42syl3anc 1184 . . . . . . . 8  |-  ( ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  (
c  e.  |^| b  /\  x  e.  |^| b
) )  /\  y  e.  b )  ->  (
c  i^i  x )  e.  y )
4443ralrimiva 2757 . . . . . . 7  |-  ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  (
c  e.  |^| b  /\  x  e.  |^| b
) )  ->  A. y  e.  b  ( c  i^i  x )  e.  y )
4526inex1 4312 . . . . . . . 8  |-  ( c  i^i  x )  e. 
_V
4645elint2 4025 . . . . . . 7  |-  ( ( c  i^i  x )  e.  |^| b  <->  A. y  e.  b  ( c  i^i  x )  e.  y )
4744, 46sylibr 204 . . . . . 6  |-  ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  (
c  e.  |^| b  /\  x  e.  |^| b
) )  ->  (
c  i^i  x )  e.  |^| b )
4847ralrimivva 2766 . . . . 5  |-  ( ( b  C_  (TopOn `  B
)  /\  b  =/=  (/) )  ->  A. c  e.  |^| b A. x  e.  |^| b ( c  i^i  x )  e. 
|^| b )
49 intex 4324 . . . . . . . 8  |-  ( b  =/=  (/)  <->  |^| b  e.  _V )
5049biimpi 187 . . . . . . 7  |-  ( b  =/=  (/)  ->  |^| b  e. 
_V )
5150adantl 453 . . . . . 6  |-  ( ( b  C_  (TopOn `  B
)  /\  b  =/=  (/) )  ->  |^| b  e. 
_V )
52 istopg 16931 . . . . . 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 889 . . . 4  |-  ( ( b  C_  (TopOn `  B
)  /\  b  =/=  (/) )  ->  |^| b  e. 
Top )
55543adant1 975 . . 3  |-  ( ( B  e.  V  /\  b  C_  (TopOn `  B
)  /\  b  =/=  (/) )  ->  |^| b  e. 
Top )
56 n0 3605 . . . . . . . . . . 11  |-  ( b  =/=  (/)  <->  E. x  x  e.  b )
5756biimpi 187 . . . . . . . . . 10  |-  ( b  =/=  (/)  ->  E. x  x  e.  b )
5857ad2antlr 708 . . . . . . . . 9  |-  ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  c  e.  |^| b )  ->  E. x  x  e.  b )
5918sselda 3316 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  b  /\  c  e.  |^| b )  ->  c  e.  x
)
6059ancoms 440 . . . . . . . . . . . . . 14  |-  ( ( c  e.  |^| b  /\  x  e.  b
)  ->  c  e.  x )
61 elssuni 4011 . . . . . . . . . . . . . 14  |-  ( c  e.  x  ->  c  C_ 
U. x )
6260, 61syl 16 . . . . . . . . . . . . 13  |-  ( ( c  e.  |^| b  /\  x  e.  b
)  ->  c  C_  U. x )
6362adantl 453 . . . . . . . . . . . 12  |-  ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  (
c  e.  |^| b  /\  x  e.  b
) )  ->  c  C_ 
U. x )
6413adantrl 697 . . . . . . . . . . . . 13  |-  ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  (
c  e.  |^| b  /\  x  e.  b
) )  ->  x  e.  (TopOn `  B )
)
65 toponuni 16955 . . . . . . . . . . . . 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 3353 . . . . . . . . . . 11  |-  ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  (
c  e.  |^| b  /\  x  e.  b
) )  ->  c  C_  B )
6867expr 599 . . . . . . . . . 10  |-  ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  c  e.  |^| b )  -> 
( x  e.  b  ->  c  C_  B
) )
6968exlimdv 1643 . . . . . . . . 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 2757 . . . . . . 7  |-  ( ( b  C_  (TopOn `  B
)  /\  b  =/=  (/) )  ->  A. c  e.  |^| b c  C_  B )
72 unissb 4013 . . . . . . 7  |-  ( U. |^| b  C_  B  <->  A. c  e.  |^| b c  C_  B )
7371, 72sylibr 204 . . . . . 6  |-  ( ( b  C_  (TopOn `  B
)  /\  b  =/=  (/) )  ->  U. |^| b  C_  B )
74733adant1 975 . . . . 5  |-  ( ( B  e.  V  /\  b  C_  (TopOn `  B
)  /\  b  =/=  (/) )  ->  U. |^| b  C_  B )
7512sselda 3316 . . . . . . . . . 10  |-  ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  c  e.  b )  ->  c  e.  (TopOn `  B )
)
76 toponuni 16955 . . . . . . . . . 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 16954 . . . . . . . . . 10  |-  ( c  e.  (TopOn `  B
)  ->  c  e.  Top )
79 eqid 2412 . . . . . . . . . . 11  |-  U. c  =  U. c
8079topopn 16942 . . . . . . . . . 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 2486 . . . . . . . 8  |-  ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  c  e.  b )  ->  B  e.  c )
8382ralrimiva 2757 . . . . . . 7  |-  ( ( b  C_  (TopOn `  B
)  /\  b  =/=  (/) )  ->  A. c  e.  b  B  e.  c )
84833adant1 975 . . . . . 6  |-  ( ( B  e.  V  /\  b  C_  (TopOn `  B
)  /\  b  =/=  (/) )  ->  A. c  e.  b  B  e.  c )
85 elintg 4026 . . . . . . 7  |-  ( B  e.  V  ->  ( B  e.  |^| b  <->  A. c  e.  b  B  e.  c ) )
86853ad2ant1 978 . . . . . 6  |-  ( ( B  e.  V  /\  b  C_  (TopOn `  B
)  /\  b  =/=  (/) )  ->  ( B  e.  |^| b  <->  A. c  e.  b  B  e.  c ) )
8784, 86mpbird 224 . . . . 5  |-  ( ( B  e.  V  /\  b  C_  (TopOn `  B
)  /\  b  =/=  (/) )  ->  B  e.  |^| b )
88 unissel 4012 . . . . 5  |-  ( ( U. |^| b  C_  B  /\  B  e.  |^| b )  ->  U. |^| b  =  B )
8974, 87, 88syl2anc 643 . . . 4  |-  ( ( B  e.  V  /\  b  C_  (TopOn `  B
)  /\  b  =/=  (/) )  ->  U. |^| b  =  B )
9089eqcomd 2417 . . 3  |-  ( ( B  e.  V  /\  b  C_  (TopOn `  B
)  /\  b  =/=  (/) )  ->  B  =  U. |^| b )
91 istopon 16953 . . 3  |-  ( |^| b  e.  (TopOn `  B
)  <->  ( |^| b  e.  Top  /\  B  = 
U. |^| b ) )
9255, 90, 91sylanbrc 646 . 2  |-  ( ( B  e.  V  /\  b  C_  (TopOn `  B
)  /\  b  =/=  (/) )  ->  |^| b  e.  (TopOn `  B )
)
9310, 11, 92ismred 13790 1  |-  ( B  e.  V  ->  (TopOn `  B )  e.  (Moore `  ~P B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936   A.wal 1546   E.wex 1547    = wceq 1649    e. wcel 1721    =/= wne 2575   A.wral 2674   _Vcvv 2924    i^i cin 3287    C_ wss 3288   (/)c0 3596   ~Pcpw 3767   U.cuni 3983   |^|cint 4018   ` cfv 5421  Moorecmre 13770   Topctop 16921  TopOnctopon 16922
This theorem is referenced by:  topmtcl  26290
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-sbc 3130  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-int 4019  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-iota 5385  df-fun 5423  df-fv 5429  df-mre 13774  df-top 16926  df-topon 16929
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