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Theorem topmeet 25725
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 25724 . . . 4  |-  ( ( X  e.  V  /\  S  C_  (TopOn `  X
) )  ->  ( ~P X  i^i  |^| S
)  e.  (TopOn `  X ) )
2 inss2 3390 . . . . . . 7  |-  ( ~P X  i^i  |^| S
)  C_  |^| S
3 intss1 3877 . . . . . . 7  |-  ( j  e.  S  ->  |^| S  C_  j )
42, 3syl5ss 3190 . . . . . 6  |-  ( j  e.  S  ->  ( ~P X  i^i  |^| S
)  C_  j )
54rgen 2608 . . . . 5  |-  A. j  e.  S  ( ~P X  i^i  |^| S )  C_  j
6 sseq1 3199 . . . . . . 7  |-  ( k  =  ( ~P X  i^i  |^| S )  -> 
( k  C_  j  <->  ( ~P X  i^i  |^| S )  C_  j
) )
76ralbidv 2563 . . . . . 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 2923 . . . . 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 3855 . . 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 16665 . . . . . . . . 9  |-  ( k  e.  (TopOn `  X
)  ->  X  =  U. k )
14 eqimss2 3231 . . . . . . . . 9  |-  ( X  =  U. k  ->  U. k  C_  X )
1513, 14syl 15 . . . . . . . 8  |-  ( k  e.  (TopOn `  X
)  ->  U. k  C_  X )
16 sspwuni 3987 . . . . . . . 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 3878 . . . . . . 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 3393 . . . . 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 2791 . . . . . 6  |-  k  e. 
_V
2423elpw 3631 . . . . 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 3253 . . 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 3987 . . 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 3196 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 1623    e. wcel 1684   A.wral 2543   {crab 2547    i^i cin 3151    C_ wss 3152   ~Pcpw 3625   U.cuni 3827   |^|cint 3862   ` cfv 5255  TopOnctopon 16632
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-int 3863  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-mre 13488  df-top 16636  df-topon 16639
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