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Theorem topmeet 26395
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 26394 . . . 4  |-  ( ( X  e.  V  /\  S  C_  (TopOn `  X
) )  ->  ( ~P X  i^i  |^| S
)  e.  (TopOn `  X ) )
2 inss2 3564 . . . . . . 7  |-  ( ~P X  i^i  |^| S
)  C_  |^| S
3 intss1 4067 . . . . . . 7  |-  ( j  e.  S  ->  |^| S  C_  j )
42, 3syl5ss 3361 . . . . . 6  |-  ( j  e.  S  ->  ( ~P X  i^i  |^| S
)  C_  j )
54rgen 2773 . . . . 5  |-  A. j  e.  S  ( ~P X  i^i  |^| S )  C_  j
6 sseq1 3371 . . . . . . 7  |-  ( k  =  ( ~P X  i^i  |^| S )  -> 
( k  C_  j  <->  ( ~P X  i^i  |^| S )  C_  j
) )
76ralbidv 2727 . . . . . 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 3094 . . . . 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 887 . . . 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 205 . . 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 4045 . . 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 16 . 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 16994 . . . . . . . . 9  |-  ( k  e.  (TopOn `  X
)  ->  X  =  U. k )
14 eqimss2 3403 . . . . . . . . 9  |-  ( X  =  U. k  ->  U. k  C_  X )
1513, 14syl 16 . . . . . . . 8  |-  ( k  e.  (TopOn `  X
)  ->  U. k  C_  X )
16 sspwuni 4178 . . . . . . . 8  |-  ( k 
C_  ~P X  <->  U. k  C_  X )
1715, 16sylibr 205 . . . . . . 7  |-  ( k  e.  (TopOn `  X
)  ->  k  C_  ~P X )
18173ad2ant2 980 . . . . . 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 960 . . . . . . 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 4068 . . . . . . 7  |-  ( k 
C_  |^| S  <->  A. j  e.  S  k  C_  j )
2119, 20sylibr 205 . . . . . 6  |-  ( ( ( X  e.  V  /\  S  C_  (TopOn `  X ) )  /\  k  e.  (TopOn `  X
)  /\  A. j  e.  S  k  C_  j )  ->  k  C_ 
|^| S )
2218, 21ssind 3567 . . . . 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 2961 . . . . . 6  |-  k  e. 
_V
2423elpw 3807 . . . . 5  |-  ( k  e.  ~P ( ~P X  i^i  |^| S
)  <->  k  C_  ( ~P X  i^i  |^| S
) )
2522, 24sylibr 205 . . . 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 3425 . . 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 4178 . . 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 190 . 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 3367 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 360    /\ w3a 937    = wceq 1653    e. wcel 1726   A.wral 2707   {crab 2711    i^i cin 3321    C_ wss 3322   ~Pcpw 3801   U.cuni 4017   |^|cint 4052   ` cfv 5456  TopOnctopon 16961
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-int 4053  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-iota 5420  df-fun 5458  df-fv 5464  df-mre 13813  df-top 16965  df-topon 16968
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