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Theorem topmeet 26416
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 26415 . . . 4  |-  ( ( X  e.  V  /\  S  C_  (TopOn `  X
) )  ->  ( ~P X  i^i  |^| S
)  e.  (TopOn `  X ) )
2 inss2 3403 . . . . . . 7  |-  ( ~P X  i^i  |^| S
)  C_  |^| S
3 intss1 3893 . . . . . . 7  |-  ( j  e.  S  ->  |^| S  C_  j )
42, 3syl5ss 3203 . . . . . 6  |-  ( j  e.  S  ->  ( ~P X  i^i  |^| S
)  C_  j )
54rgen 2621 . . . . 5  |-  A. j  e.  S  ( ~P X  i^i  |^| S )  C_  j
6 sseq1 3212 . . . . . . 7  |-  ( k  =  ( ~P X  i^i  |^| S )  -> 
( k  C_  j  <->  ( ~P X  i^i  |^| S )  C_  j
) )
76ralbidv 2576 . . . . . 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 2936 . . . . 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 885 . . . 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 203 . . 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 3871 . . 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 15 . 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 16681 . . . . . . . . 9  |-  ( k  e.  (TopOn `  X
)  ->  X  =  U. k )
14 eqimss2 3244 . . . . . . . . 9  |-  ( X  =  U. k  ->  U. k  C_  X )
1513, 14syl 15 . . . . . . . 8  |-  ( k  e.  (TopOn `  X
)  ->  U. k  C_  X )
16 sspwuni 4003 . . . . . . . 8  |-  ( k 
C_  ~P X  <->  U. k  C_  X )
1715, 16sylibr 203 . . . . . . 7  |-  ( k  e.  (TopOn `  X
)  ->  k  C_  ~P X )
18173ad2ant2 977 . . . . . 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 957 . . . . . . 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 3894 . . . . . . 7  |-  ( k 
C_  |^| S  <->  A. j  e.  S  k  C_  j )
2119, 20sylibr 203 . . . . . 6  |-  ( ( ( X  e.  V  /\  S  C_  (TopOn `  X ) )  /\  k  e.  (TopOn `  X
)  /\  A. j  e.  S  k  C_  j )  ->  k  C_ 
|^| S )
2218, 21ssind 3406 . . . . 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 vex 2804 . . . . . 6  |-  k  e. 
_V
2423elpw 3644 . . . . 5  |-  ( k  e.  ~P ( ~P X  i^i  |^| S
)  <->  k  C_  ( ~P X  i^i  |^| S
) )
2522, 24sylibr 203 . . . 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 ) )
2625rabssdv 3266 . . 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
) )
27 sspwuni 4003 . . 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
) )
2826, 27sylib 188 . 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
) )
2912, 28eqssd 3209 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 set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   A.wral 2556   {crab 2560    i^i cin 3164    C_ wss 3165   ~Pcpw 3638   U.cuni 3843   |^|cint 3878   ` cfv 5271  TopOnctopon 16648
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-int 3879  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279  df-mre 13504  df-top 16652  df-topon 16655
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