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Theorem toponmre 18813
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 18716. (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 18648 . . . . . 6  |-  ( b  e.  (TopOn `  B
)  ->  B  =  U. b )
2 eqimss2 3507 . . . . . . 7  |-  ( B  =  U. b  ->  U. b  C_  B )
3 sspwuni 4354 . . . . . . 7  |-  ( b 
C_  ~P B  <->  U. b  C_  B )
42, 3sylibr 212 . . . . . 6  |-  ( B  =  U. b  -> 
b  C_  ~P B
)
51, 4syl 16 . . . . 5  |-  ( b  e.  (TopOn `  B
)  ->  b  C_  ~P B )
6 selpw 3965 . . . . 5  |-  ( b  e.  ~P ~P B  <->  b 
C_  ~P B )
75, 6sylibr 212 . . . 4  |-  ( b  e.  (TopOn `  B
)  ->  b  e.  ~P ~P B )
87ssriv 3458 . . 3  |-  (TopOn `  B )  C_  ~P ~P B
98a1i 11 . 2  |-  ( B  e.  V  ->  (TopOn `  B )  C_  ~P ~P B )
10 distopon 18717 . 2  |-  ( B  e.  V  ->  ~P B  e.  (TopOn `  B
) )
11 simpl 457 . . . . . . . . . . . . . 14  |-  ( ( b  C_  (TopOn `  B
)  /\  b  =/=  (/) )  ->  b  C_  (TopOn `  B ) )
1211sselda 3454 . . . . . . . . . . . . 13  |-  ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  x  e.  b )  ->  x  e.  (TopOn `  B )
)
1312adantrl 715 . . . . . . . . . . . 12  |-  ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  (
c  C_  |^| b  /\  x  e.  b )
)  ->  x  e.  (TopOn `  B ) )
14 topontop 18647 . . . . . . . . . . . 12  |-  ( x  e.  (TopOn `  B
)  ->  x  e.  Top )
1513, 14syl 16 . . . . . . . . . . 11  |-  ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  (
c  C_  |^| b  /\  x  e.  b )
)  ->  x  e.  Top )
16 simpl 457 . . . . . . . . . . . . 13  |-  ( ( c  C_  |^| b  /\  x  e.  b )  ->  c  C_  |^| b )
17 intss1 4241 . . . . . . . . . . . . . 14  |-  ( x  e.  b  ->  |^| b  C_  x )
1817adantl 466 . . . . . . . . . . . . 13  |-  ( ( c  C_  |^| b  /\  x  e.  b )  ->  |^| b  C_  x
)
1916, 18sstrd 3464 . . . . . . . . . . . 12  |-  ( ( c  C_  |^| b  /\  x  e.  b )  ->  c  C_  x )
2019adantl 466 . . . . . . . . . . 11  |-  ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  (
c  C_  |^| b  /\  x  e.  b )
)  ->  c  C_  x )
21 uniopn 18626 . . . . . . . . . . 11  |-  ( ( x  e.  Top  /\  c  C_  x )  ->  U. c  e.  x
)
2215, 20, 21syl2anc 661 . . . . . . . . . 10  |-  ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  (
c  C_  |^| b  /\  x  e.  b )
)  ->  U. c  e.  x )
2322expr 615 . . . . . . . . 9  |-  ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  c  C_ 
|^| b )  -> 
( x  e.  b  ->  U. c  e.  x
) )
2423ralrimiv 2820 . . . . . . . 8  |-  ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  c  C_ 
|^| b )  ->  A. x  e.  b  U. c  e.  x
)
25 vex 3071 . . . . . . . . . 10  |-  c  e. 
_V
2625uniex 6476 . . . . . . . . 9  |-  U. c  e.  _V
2726elint2 4233 . . . . . . . 8  |-  ( U. c  e.  |^| b  <->  A. x  e.  b  U. c  e.  x )
2824, 27sylibr 212 . . . . . . 7  |-  ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  c  C_ 
|^| b )  ->  U. c  e.  |^| b
)
2928ex 434 . . . . . 6  |-  ( ( b  C_  (TopOn `  B
)  /\  b  =/=  (/) )  ->  ( c  C_ 
|^| b  ->  U. c  e.  |^| b ) )
3029alrimiv 1686 . . . . 5  |-  ( ( b  C_  (TopOn `  B
)  /\  b  =/=  (/) )  ->  A. c
( c  C_  |^| b  ->  U. c  e.  |^| b ) )
31 simpll 753 . . . . . . . . . . 11  |-  ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  (
c  e.  |^| b  /\  x  e.  |^| b
) )  ->  b  C_  (TopOn `  B )
)
3231sselda 3454 . . . . . . . . . 10  |-  ( ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  (
c  e.  |^| b  /\  x  e.  |^| b
) )  /\  y  e.  b )  ->  y  e.  (TopOn `  B )
)
33 topontop 18647 . . . . . . . . . 10  |-  ( y  e.  (TopOn `  B
)  ->  y  e.  Top )
3432, 33syl 16 . . . . . . . . 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 466 . . . . . . . . . 10  |-  ( ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  (
c  e.  |^| b  /\  x  e.  |^| b
) )  /\  y  e.  b )  ->  |^| b  C_  y )
37 simplrl 759 . . . . . . . . . 10  |-  ( ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  (
c  e.  |^| b  /\  x  e.  |^| b
) )  /\  y  e.  b )  ->  c  e.  |^| b )
3836, 37sseldd 3455 . . . . . . . . 9  |-  ( ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  (
c  e.  |^| b  /\  x  e.  |^| b
) )  /\  y  e.  b )  ->  c  e.  y )
39 simplrr 760 . . . . . . . . . 10  |-  ( ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  (
c  e.  |^| b  /\  x  e.  |^| b
) )  /\  y  e.  b )  ->  x  e.  |^| b )
4036, 39sseldd 3455 . . . . . . . . 9  |-  ( ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  (
c  e.  |^| b  /\  x  e.  |^| b
) )  /\  y  e.  b )  ->  x  e.  y )
41 inopn 18628 . . . . . . . . 9  |-  ( ( y  e.  Top  /\  c  e.  y  /\  x  e.  y )  ->  ( c  i^i  x
)  e.  y )
4234, 38, 40, 41syl3anc 1219 . . . . . . . 8  |-  ( ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  (
c  e.  |^| b  /\  x  e.  |^| b
) )  /\  y  e.  b )  ->  (
c  i^i  x )  e.  y )
4342ralrimiva 2822 . . . . . . 7  |-  ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  (
c  e.  |^| b  /\  x  e.  |^| b
) )  ->  A. y  e.  b  ( c  i^i  x )  e.  y )
4425inex1 4531 . . . . . . . 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 212 . . . . . 6  |-  ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  (
c  e.  |^| b  /\  x  e.  |^| b
) )  ->  (
c  i^i  x )  e.  |^| b )
4746ralrimivva 2904 . . . . 5  |-  ( ( b  C_  (TopOn `  B
)  /\  b  =/=  (/) )  ->  A. c  e.  |^| b A. x  e.  |^| b ( c  i^i  x )  e. 
|^| b )
48 intex 4546 . . . . . . . 8  |-  ( b  =/=  (/)  <->  |^| b  e.  _V )
4948biimpi 194 . . . . . . 7  |-  ( b  =/=  (/)  ->  |^| b  e. 
_V )
5049adantl 466 . . . . . 6  |-  ( ( b  C_  (TopOn `  B
)  /\  b  =/=  (/) )  ->  |^| b  e. 
_V )
51 istopg 18624 . . . . . 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 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
) ) )
5330, 47, 52mpbir2and 913 . . . 4  |-  ( ( b  C_  (TopOn `  B
)  /\  b  =/=  (/) )  ->  |^| b  e. 
Top )
54533adant1 1006 . . 3  |-  ( ( B  e.  V  /\  b  C_  (TopOn `  B
)  /\  b  =/=  (/) )  ->  |^| b  e. 
Top )
55 n0 3744 . . . . . . . . . . 11  |-  ( b  =/=  (/)  <->  E. x  x  e.  b )
5655biimpi 194 . . . . . . . . . 10  |-  ( b  =/=  (/)  ->  E. x  x  e.  b )
5756ad2antlr 726 . . . . . . . . 9  |-  ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  c  e.  |^| b )  ->  E. x  x  e.  b )
5817sselda 3454 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  b  /\  c  e.  |^| b )  ->  c  e.  x
)
5958ancoms 453 . . . . . . . . . . . . . 14  |-  ( ( c  e.  |^| b  /\  x  e.  b
)  ->  c  e.  x )
60 elssuni 4219 . . . . . . . . . . . . . 14  |-  ( c  e.  x  ->  c  C_ 
U. x )
6159, 60syl 16 . . . . . . . . . . . . 13  |-  ( ( c  e.  |^| b  /\  x  e.  b
)  ->  c  C_  U. x )
6261adantl 466 . . . . . . . . . . . 12  |-  ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  (
c  e.  |^| b  /\  x  e.  b
) )  ->  c  C_ 
U. x )
6312adantrl 715 . . . . . . . . . . . . 13  |-  ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  (
c  e.  |^| b  /\  x  e.  b
) )  ->  x  e.  (TopOn `  B )
)
64 toponuni 18648 . . . . . . . . . . . . 13  |-  ( x  e.  (TopOn `  B
)  ->  B  =  U. x )
6563, 64syl 16 . . . . . . . . . . . 12  |-  ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  (
c  e.  |^| b  /\  x  e.  b
) )  ->  B  =  U. x )
6662, 65sseqtr4d 3491 . . . . . . . . . . 11  |-  ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  (
c  e.  |^| b  /\  x  e.  b
) )  ->  c  C_  B )
6766expr 615 . . . . . . . . . 10  |-  ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  c  e.  |^| b )  -> 
( x  e.  b  ->  c  C_  B
) )
6867exlimdv 1691 . . . . . . . . 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 2822 . . . . . . 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 212 . . . . . 6  |-  ( ( b  C_  (TopOn `  B
)  /\  b  =/=  (/) )  ->  U. |^| b  C_  B )
73723adant1 1006 . . . . 5  |-  ( ( B  e.  V  /\  b  C_  (TopOn `  B
)  /\  b  =/=  (/) )  ->  U. |^| b  C_  B )
7411sselda 3454 . . . . . . . . . 10  |-  ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  c  e.  b )  ->  c  e.  (TopOn `  B )
)
75 toponuni 18648 . . . . . . . . . 10  |-  ( c  e.  (TopOn `  B
)  ->  B  =  U. c )
7674, 75syl 16 . . . . . . . . 9  |-  ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  c  e.  b )  ->  B  =  U. c )
77 topontop 18647 . . . . . . . . . 10  |-  ( c  e.  (TopOn `  B
)  ->  c  e.  Top )
78 eqid 2451 . . . . . . . . . . 11  |-  U. c  =  U. c
7978topopn 18635 . . . . . . . . . 10  |-  ( c  e.  Top  ->  U. c  e.  c )
8074, 77, 793syl 20 . . . . . . . . 9  |-  ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  c  e.  b )  ->  U. c  e.  c )
8176, 80eqeltrd 2539 . . . . . . . 8  |-  ( ( ( b  C_  (TopOn `  B )  /\  b  =/=  (/) )  /\  c  e.  b )  ->  B  e.  c )
8281ralrimiva 2822 . . . . . . 7  |-  ( ( b  C_  (TopOn `  B
)  /\  b  =/=  (/) )  ->  A. c  e.  b  B  e.  c )
83823adant1 1006 . . . . . 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 1009 . . . . . 6  |-  ( ( B  e.  V  /\  b  C_  (TopOn `  B
)  /\  b  =/=  (/) )  ->  ( B  e.  |^| b  <->  A. c  e.  b  B  e.  c ) )
8683, 85mpbird 232 . . . . 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 661 . . . 4  |-  ( ( B  e.  V  /\  b  C_  (TopOn `  B
)  /\  b  =/=  (/) )  ->  U. |^| b  =  B )
8988eqcomd 2459 . . 3  |-  ( ( B  e.  V  /\  b  C_  (TopOn `  B
)  /\  b  =/=  (/) )  ->  B  =  U. |^| b )
90 istopon 18646 . . 3  |-  ( |^| b  e.  (TopOn `  B
)  <->  ( |^| b  e.  Top  /\  B  = 
U. |^| b ) )
9154, 89, 90sylanbrc 664 . 2  |-  ( ( B  e.  V  /\  b  C_  (TopOn `  B
)  /\  b  =/=  (/) )  ->  |^| b  e.  (TopOn `  B )
)
929, 10, 91ismred 14642 1  |-  ( B  e.  V  ->  (TopOn `  B )  e.  (Moore `  ~P B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965   A.wal 1368    = wceq 1370   E.wex 1587    e. wcel 1758    =/= wne 2644   A.wral 2795   _Vcvv 3068    i^i cin 3425    C_ wss 3426   (/)c0 3735   ~Pcpw 3958   U.cuni 4189   |^|cint 4226   ` cfv 5516  Moorecmre 14622   Topctop 18614  TopOnctopon 18615
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3070  df-sbc 3285  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-op 3982  df-uni 4190  df-int 4227  df-br 4391  df-opab 4449  df-mpt 4450  df-id 4734  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-iota 5479  df-fun 5518  df-fv 5524  df-mre 14626  df-top 18619  df-topon 18622
This theorem is referenced by:  topmtcl  28722
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