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Theorem toponmre 20186
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 20088. (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 20019 . . . . . 6  |-  ( b  e.  (TopOn `  B
)  ->  B  =  U. b )
2 eqimss2 3471 . . . . . . 7  |-  ( B  =  U. b  ->  U. b  C_  B )
3 sspwuni 4360 . . . . . . 7  |-  ( b 
C_  ~P B  <->  U. b  C_  B )
42, 3sylibr 217 . . . . . 6  |-  ( B  =  U. b  -> 
b  C_  ~P B
)
51, 4syl 17 . . . . 5  |-  ( b  e.  (TopOn `  B
)  ->  b  C_  ~P B )
6 selpw 3949 . . . . 5  |-  ( b  e.  ~P ~P B  <->  b 
C_  ~P B )
75, 6sylibr 217 . . . 4  |-  ( b  e.  (TopOn `  B
)  ->  b  e.  ~P ~P B )
87ssriv 3422 . . 3  |-  (TopOn `  B )  C_  ~P ~P B
98a1i 11 . 2  |-  ( B  e.  V  ->  (TopOn `  B )  C_  ~P ~P B )
10 distopon 20089 . 2  |-  ( B  e.  V  ->  ~P B  e.  (TopOn `  B
) )
11 simpl 464 . . . . . . . . . . . . . 14  |-  ( ( b  C_  (TopOn `  B
)  /\  b  =/=  (/) )  ->  b  C_  (TopOn `  B ) )
1211sselda 3418 . . . . . . . . . . . . 13  |-  ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  x  e.  b )  ->  x  e.  (TopOn `  B )
)
1312adantrl 730 . . . . . . . . . . . 12  |-  ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  (
c  C_  |^| b  /\  x  e.  b )
)  ->  x  e.  (TopOn `  B ) )
14 topontop 20018 . . . . . . . . . . . 12  |-  ( x  e.  (TopOn `  B
)  ->  x  e.  Top )
1513, 14syl 17 . . . . . . . . . . 11  |-  ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  (
c  C_  |^| b  /\  x  e.  b )
)  ->  x  e.  Top )
16 simpl 464 . . . . . . . . . . . . 13  |-  ( ( c  C_  |^| b  /\  x  e.  b )  ->  c  C_  |^| b )
17 intss1 4241 . . . . . . . . . . . . . 14  |-  ( x  e.  b  ->  |^| b  C_  x )
1817adantl 473 . . . . . . . . . . . . 13  |-  ( ( c  C_  |^| b  /\  x  e.  b )  ->  |^| b  C_  x
)
1916, 18sstrd 3428 . . . . . . . . . . . 12  |-  ( ( c  C_  |^| b  /\  x  e.  b )  ->  c  C_  x )
2019adantl 473 . . . . . . . . . . 11  |-  ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  (
c  C_  |^| b  /\  x  e.  b )
)  ->  c  C_  x )
21 uniopn 20004 . . . . . . . . . . 11  |-  ( ( x  e.  Top  /\  c  C_  x )  ->  U. c  e.  x
)
2215, 20, 21syl2anc 673 . . . . . . . . . 10  |-  ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  (
c  C_  |^| b  /\  x  e.  b )
)  ->  U. c  e.  x )
2322expr 626 . . . . . . . . 9  |-  ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  c  C_ 
|^| b )  -> 
( x  e.  b  ->  U. c  e.  x
) )
2423ralrimiv 2808 . . . . . . . 8  |-  ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  c  C_ 
|^| b )  ->  A. x  e.  b  U. c  e.  x
)
25 vex 3034 . . . . . . . . . 10  |-  c  e. 
_V
2625uniex 6606 . . . . . . . . 9  |-  U. c  e.  _V
2726elint2 4233 . . . . . . . 8  |-  ( U. c  e.  |^| b  <->  A. x  e.  b  U. c  e.  x )
2824, 27sylibr 217 . . . . . . 7  |-  ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  c  C_ 
|^| b )  ->  U. c  e.  |^| b
)
2928ex 441 . . . . . 6  |-  ( ( b  C_  (TopOn `  B
)  /\  b  =/=  (/) )  ->  ( c  C_ 
|^| b  ->  U. c  e.  |^| b ) )
3029alrimiv 1781 . . . . 5  |-  ( ( b  C_  (TopOn `  B
)  /\  b  =/=  (/) )  ->  A. c
( c  C_  |^| b  ->  U. c  e.  |^| b ) )
31 simpll 768 . . . . . . . . . . 11  |-  ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  (
c  e.  |^| b  /\  x  e.  |^| b
) )  ->  b  C_  (TopOn `  B )
)
3231sselda 3418 . . . . . . . . . 10  |-  ( ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  (
c  e.  |^| b  /\  x  e.  |^| b
) )  /\  y  e.  b )  ->  y  e.  (TopOn `  B )
)
33 topontop 20018 . . . . . . . . . 10  |-  ( y  e.  (TopOn `  B
)  ->  y  e.  Top )
3432, 33syl 17 . . . . . . . . 9  |-  ( ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  (
c  e.  |^| b  /\  x  e.  |^| b
) )  /\  y  e.  b )  ->  y  e.  Top )
35 intss1 4241 . . . . . . . . . . 11  |-  ( y  e.  b  ->  |^| b  C_  y )
3635adantl 473 . . . . . . . . . 10  |-  ( ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  (
c  e.  |^| b  /\  x  e.  |^| b
) )  /\  y  e.  b )  ->  |^| b  C_  y )
37 simplrl 778 . . . . . . . . . 10  |-  ( ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  (
c  e.  |^| b  /\  x  e.  |^| b
) )  /\  y  e.  b )  ->  c  e.  |^| b )
3836, 37sseldd 3419 . . . . . . . . 9  |-  ( ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  (
c  e.  |^| b  /\  x  e.  |^| b
) )  /\  y  e.  b )  ->  c  e.  y )
39 simplrr 779 . . . . . . . . . 10  |-  ( ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  (
c  e.  |^| b  /\  x  e.  |^| b
) )  /\  y  e.  b )  ->  x  e.  |^| b )
4036, 39sseldd 3419 . . . . . . . . 9  |-  ( ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  (
c  e.  |^| b  /\  x  e.  |^| b
) )  /\  y  e.  b )  ->  x  e.  y )
41 inopn 20006 . . . . . . . . 9  |-  ( ( y  e.  Top  /\  c  e.  y  /\  x  e.  y )  ->  ( c  i^i  x
)  e.  y )
4234, 38, 40, 41syl3anc 1292 . . . . . . . 8  |-  ( ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  (
c  e.  |^| b  /\  x  e.  |^| b
) )  /\  y  e.  b )  ->  (
c  i^i  x )  e.  y )
4342ralrimiva 2809 . . . . . . 7  |-  ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  (
c  e.  |^| b  /\  x  e.  |^| b
) )  ->  A. y  e.  b  ( c  i^i  x )  e.  y )
4425inex1 4537 . . . . . . . 8  |-  ( c  i^i  x )  e. 
_V
4544elint2 4233 . . . . . . 7  |-  ( ( c  i^i  x )  e.  |^| b  <->  A. y  e.  b  ( c  i^i  x )  e.  y )
4643, 45sylibr 217 . . . . . 6  |-  ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  (
c  e.  |^| b  /\  x  e.  |^| b
) )  ->  (
c  i^i  x )  e.  |^| b )
4746ralrimivva 2814 . . . . 5  |-  ( ( b  C_  (TopOn `  B
)  /\  b  =/=  (/) )  ->  A. c  e.  |^| b A. x  e.  |^| b ( c  i^i  x )  e. 
|^| b )
48 intex 4557 . . . . . . . 8  |-  ( b  =/=  (/)  <->  |^| b  e.  _V )
4948biimpi 199 . . . . . . 7  |-  ( b  =/=  (/)  ->  |^| b  e. 
_V )
5049adantl 473 . . . . . 6  |-  ( ( b  C_  (TopOn `  B
)  /\  b  =/=  (/) )  ->  |^| b  e. 
_V )
51 istopg 20002 . . . . . 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
) ) )
5250, 51syl 17 . . . . 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
) ) )
5330, 47, 52mpbir2and 936 . . . 4  |-  ( ( b  C_  (TopOn `  B
)  /\  b  =/=  (/) )  ->  |^| b  e. 
Top )
54533adant1 1048 . . 3  |-  ( ( B  e.  V  /\  b  C_  (TopOn `  B
)  /\  b  =/=  (/) )  ->  |^| b  e. 
Top )
55 n0 3732 . . . . . . . . . . 11  |-  ( b  =/=  (/)  <->  E. x  x  e.  b )
5655biimpi 199 . . . . . . . . . 10  |-  ( b  =/=  (/)  ->  E. x  x  e.  b )
5756ad2antlr 741 . . . . . . . . 9  |-  ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  c  e.  |^| b )  ->  E. x  x  e.  b )
5817sselda 3418 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  b  /\  c  e.  |^| b )  ->  c  e.  x
)
5958ancoms 460 . . . . . . . . . . . . . 14  |-  ( ( c  e.  |^| b  /\  x  e.  b
)  ->  c  e.  x )
60 elssuni 4219 . . . . . . . . . . . . . 14  |-  ( c  e.  x  ->  c  C_ 
U. x )
6159, 60syl 17 . . . . . . . . . . . . 13  |-  ( ( c  e.  |^| b  /\  x  e.  b
)  ->  c  C_  U. x )
6261adantl 473 . . . . . . . . . . . 12  |-  ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  (
c  e.  |^| b  /\  x  e.  b
) )  ->  c  C_ 
U. x )
6312adantrl 730 . . . . . . . . . . . . 13  |-  ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  (
c  e.  |^| b  /\  x  e.  b
) )  ->  x  e.  (TopOn `  B )
)
64 toponuni 20019 . . . . . . . . . . . . 13  |-  ( x  e.  (TopOn `  B
)  ->  B  =  U. x )
6563, 64syl 17 . . . . . . . . . . . 12  |-  ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  (
c  e.  |^| b  /\  x  e.  b
) )  ->  B  =  U. x )
6662, 65sseqtr4d 3455 . . . . . . . . . . 11  |-  ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  (
c  e.  |^| b  /\  x  e.  b
) )  ->  c  C_  B )
6766expr 626 . . . . . . . . . 10  |-  ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  c  e.  |^| b )  -> 
( x  e.  b  ->  c  C_  B
) )
6867exlimdv 1787 . . . . . . . . 9  |-  ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  c  e.  |^| b )  -> 
( E. x  x  e.  b  ->  c  C_  B ) )
6957, 68mpd 15 . . . . . . . 8  |-  ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  c  e.  |^| b )  -> 
c  C_  B )
7069ralrimiva 2809 . . . . . . 7  |-  ( ( b  C_  (TopOn `  B
)  /\  b  =/=  (/) )  ->  A. c  e.  |^| b c  C_  B )
71 unissb 4221 . . . . . . 7  |-  ( U. |^| b  C_  B  <->  A. c  e.  |^| b c  C_  B )
7270, 71sylibr 217 . . . . . 6  |-  ( ( b  C_  (TopOn `  B
)  /\  b  =/=  (/) )  ->  U. |^| b  C_  B )
73723adant1 1048 . . . . 5  |-  ( ( B  e.  V  /\  b  C_  (TopOn `  B
)  /\  b  =/=  (/) )  ->  U. |^| b  C_  B )
7411sselda 3418 . . . . . . . . . 10  |-  ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  c  e.  b )  ->  c  e.  (TopOn `  B )
)
75 toponuni 20019 . . . . . . . . . 10  |-  ( c  e.  (TopOn `  B
)  ->  B  =  U. c )
7674, 75syl 17 . . . . . . . . 9  |-  ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  c  e.  b )  ->  B  =  U. c )
77 topontop 20018 . . . . . . . . . 10  |-  ( c  e.  (TopOn `  B
)  ->  c  e.  Top )
78 eqid 2471 . . . . . . . . . . 11  |-  U. c  =  U. c
7978topopn 20013 . . . . . . . . . 10  |-  ( c  e.  Top  ->  U. c  e.  c )
8074, 77, 793syl 18 . . . . . . . . 9  |-  ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  c  e.  b )  ->  U. c  e.  c )
8176, 80eqeltrd 2549 . . . . . . . 8  |-  ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  c  e.  b )  ->  B  e.  c )
8281ralrimiva 2809 . . . . . . 7  |-  ( ( b  C_  (TopOn `  B
)  /\  b  =/=  (/) )  ->  A. c  e.  b  B  e.  c )
83823adant1 1048 . . . . . 6  |-  ( ( B  e.  V  /\  b  C_  (TopOn `  B
)  /\  b  =/=  (/) )  ->  A. c  e.  b  B  e.  c )
84 elintg 4234 . . . . . . 7  |-  ( B  e.  V  ->  ( B  e.  |^| b  <->  A. c  e.  b  B  e.  c ) )
85843ad2ant1 1051 . . . . . 6  |-  ( ( B  e.  V  /\  b  C_  (TopOn `  B
)  /\  b  =/=  (/) )  ->  ( B  e.  |^| b  <->  A. c  e.  b  B  e.  c ) )
8683, 85mpbird 240 . . . . 5  |-  ( ( B  e.  V  /\  b  C_  (TopOn `  B
)  /\  b  =/=  (/) )  ->  B  e.  |^| b )
87 unissel 4220 . . . . 5  |-  ( ( U. |^| b  C_  B  /\  B  e.  |^| b )  ->  U. |^| b  =  B )
8873, 86, 87syl2anc 673 . . . 4  |-  ( ( B  e.  V  /\  b  C_  (TopOn `  B
)  /\  b  =/=  (/) )  ->  U. |^| b  =  B )
8988eqcomd 2477 . . 3  |-  ( ( B  e.  V  /\  b  C_  (TopOn `  B
)  /\  b  =/=  (/) )  ->  B  =  U. |^| b )
90 istopon 20017 . . 3  |-  ( |^| b  e.  (TopOn `  B
)  <->  ( |^| b  e.  Top  /\  B  = 
U. |^| b ) )
9154, 89, 90sylanbrc 677 . 2  |-  ( ( B  e.  V  /\  b  C_  (TopOn `  B
)  /\  b  =/=  (/) )  ->  |^| b  e.  (TopOn `  B )
)
929, 10, 91ismred 15586 1  |-  ( B  e.  V  ->  (TopOn `  B )  e.  (Moore `  ~P B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 376    /\ w3a 1007   A.wal 1450    = wceq 1452   E.wex 1671    e. wcel 1904    =/= wne 2641   A.wral 2756   _Vcvv 3031    i^i cin 3389    C_ wss 3390   (/)c0 3722   ~Pcpw 3942   U.cuni 4190   |^|cint 4226   ` cfv 5589  Moorecmre 15566   Topctop 19994  TopOnctopon 19995
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-int 4227  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-iota 5553  df-fun 5591  df-fv 5597  df-mre 15570  df-top 19998  df-topon 20000
This theorem is referenced by:  topmtcl  31090
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