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Theorem topmeet 31091
Description: Two equivalent formulations of the meet of a collection of topologies. (Contributed by Jeff Hankins, 4-Oct-2009.) (Proof shortened by Mario Carneiro, 12-Sep-2015.)
Assertion
Ref Expression
topmeet  |-  ( ( X  e.  V  /\  S  C_  (TopOn `  X
) )  ->  ( ~P X  i^i  |^| S
)  =  U. {
k  e.  (TopOn `  X )  |  A. j  e.  S  k  C_  j } )
Distinct variable groups:    j, k, S    j, V, k    j, X, k

Proof of Theorem topmeet
StepHypRef Expression
1 topmtcl 31090 . . . 4  |-  ( ( X  e.  V  /\  S  C_  (TopOn `  X
) )  ->  ( ~P X  i^i  |^| S
)  e.  (TopOn `  X ) )
2 inss2 3644 . . . . . . 7  |-  ( ~P X  i^i  |^| S
)  C_  |^| S
3 intss1 4241 . . . . . . 7  |-  ( j  e.  S  ->  |^| S  C_  j )
42, 3syl5ss 3429 . . . . . 6  |-  ( j  e.  S  ->  ( ~P X  i^i  |^| S
)  C_  j )
54rgen 2766 . . . . 5  |-  A. j  e.  S  ( ~P X  i^i  |^| S )  C_  j
6 sseq1 3439 . . . . . . 7  |-  ( k  =  ( ~P X  i^i  |^| S )  -> 
( k  C_  j  <->  ( ~P X  i^i  |^| S )  C_  j
) )
76ralbidv 2829 . . . . . 6  |-  ( k  =  ( ~P X  i^i  |^| S )  -> 
( A. j  e.  S  k  C_  j  <->  A. j  e.  S  ( ~P X  i^i  |^| S )  C_  j
) )
87elrab 3184 . . . . 5  |-  ( ( ~P X  i^i  |^| S )  e.  {
k  e.  (TopOn `  X )  |  A. j  e.  S  k  C_  j }  <->  ( ( ~P X  i^i  |^| S
)  e.  (TopOn `  X )  /\  A. j  e.  S  ( ~P X  i^i  |^| S
)  C_  j )
)
95, 8mpbiran2 933 . . . 4  |-  ( ( ~P X  i^i  |^| S )  e.  {
k  e.  (TopOn `  X )  |  A. j  e.  S  k  C_  j }  <->  ( ~P X  i^i  |^| S )  e.  (TopOn `  X )
)
101, 9sylibr 217 . . 3  |-  ( ( X  e.  V  /\  S  C_  (TopOn `  X
) )  ->  ( ~P X  i^i  |^| S
)  e.  { k  e.  (TopOn `  X
)  |  A. j  e.  S  k  C_  j } )
11 elssuni 4219 . . 3  |-  ( ( ~P X  i^i  |^| S )  e.  {
k  e.  (TopOn `  X )  |  A. j  e.  S  k  C_  j }  ->  ( ~P X  i^i  |^| S
)  C_  U. { k  e.  (TopOn `  X
)  |  A. j  e.  S  k  C_  j } )
1210, 11syl 17 . 2  |-  ( ( X  e.  V  /\  S  C_  (TopOn `  X
) )  ->  ( ~P X  i^i  |^| S
)  C_  U. { k  e.  (TopOn `  X
)  |  A. j  e.  S  k  C_  j } )
13 toponuni 20019 . . . . . . . . 9  |-  ( k  e.  (TopOn `  X
)  ->  X  =  U. k )
14 eqimss2 3471 . . . . . . . . 9  |-  ( X  =  U. k  ->  U. k  C_  X )
1513, 14syl 17 . . . . . . . 8  |-  ( k  e.  (TopOn `  X
)  ->  U. k  C_  X )
16 sspwuni 4360 . . . . . . . 8  |-  ( k 
C_  ~P X  <->  U. k  C_  X )
1715, 16sylibr 217 . . . . . . 7  |-  ( k  e.  (TopOn `  X
)  ->  k  C_  ~P X )
18173ad2ant2 1052 . . . . . 6  |-  ( ( ( X  e.  V  /\  S  C_  (TopOn `  X ) )  /\  k  e.  (TopOn `  X
)  /\  A. j  e.  S  k  C_  j )  ->  k  C_ 
~P X )
19 simp3 1032 . . . . . . 7  |-  ( ( ( X  e.  V  /\  S  C_  (TopOn `  X ) )  /\  k  e.  (TopOn `  X
)  /\  A. j  e.  S  k  C_  j )  ->  A. j  e.  S  k  C_  j )
20 ssint 4242 . . . . . . 7  |-  ( k 
C_  |^| S  <->  A. j  e.  S  k  C_  j )
2119, 20sylibr 217 . . . . . 6  |-  ( ( ( X  e.  V  /\  S  C_  (TopOn `  X ) )  /\  k  e.  (TopOn `  X
)  /\  A. j  e.  S  k  C_  j )  ->  k  C_ 
|^| S )
2218, 21ssind 3647 . . . . 5  |-  ( ( ( X  e.  V  /\  S  C_  (TopOn `  X ) )  /\  k  e.  (TopOn `  X
)  /\  A. j  e.  S  k  C_  j )  ->  k  C_  ( ~P X  i^i  |^| S ) )
23 selpw 3949 . . . . 5  |-  ( k  e.  ~P ( ~P X  i^i  |^| S
)  <->  k  C_  ( ~P X  i^i  |^| S
) )
2422, 23sylibr 217 . . . 4  |-  ( ( ( X  e.  V  /\  S  C_  (TopOn `  X ) )  /\  k  e.  (TopOn `  X
)  /\  A. j  e.  S  k  C_  j )  ->  k  e.  ~P ( ~P X  i^i  |^| S ) )
2524rabssdv 3495 . . 3  |-  ( ( X  e.  V  /\  S  C_  (TopOn `  X
) )  ->  { k  e.  (TopOn `  X
)  |  A. j  e.  S  k  C_  j }  C_  ~P ( ~P X  i^i  |^| S
) )
26 sspwuni 4360 . . 3  |-  ( { k  e.  (TopOn `  X )  |  A. j  e.  S  k  C_  j }  C_  ~P ( ~P X  i^i  |^| S )  <->  U. { k  e.  (TopOn `  X
)  |  A. j  e.  S  k  C_  j }  C_  ( ~P X  i^i  |^| S
) )
2725, 26sylib 201 . 2  |-  ( ( X  e.  V  /\  S  C_  (TopOn `  X
) )  ->  U. {
k  e.  (TopOn `  X )  |  A. j  e.  S  k  C_  j }  C_  ( ~P X  i^i  |^| S
) )
2812, 27eqssd 3435 1  |-  ( ( X  e.  V  /\  S  C_  (TopOn `  X
) )  ->  ( ~P X  i^i  |^| S
)  =  U. {
k  e.  (TopOn `  X )  |  A. j  e.  S  k  C_  j } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904   A.wral 2756   {crab 2760    i^i cin 3389    C_ wss 3390   ~Pcpw 3942   U.cuni 4190   |^|cint 4226   ` cfv 5589  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: (None)
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