Users' Mathboxes Mathbox for Jeff Hankins < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  finlocfin Unicode version

Theorem finlocfin 26299
Description: A finite cover of a topological space is a locally finite cover. (Contributed by Jeff Hankins, 21-Jan-2010.)
Hypotheses
Ref Expression
finlocfin.1  |-  X  = 
U. J
finlocfin.2  |-  Y  = 
U. A
Assertion
Ref Expression
finlocfin  |-  ( ( J  e.  Top  /\  A  e.  Fin  /\  X  =  Y )  ->  A  e.  ( LocFin `  J )
)

Proof of Theorem finlocfin
Dummy variables  n  s  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp1 955 . 2  |-  ( ( J  e.  Top  /\  A  e.  Fin  /\  X  =  Y )  ->  J  e.  Top )
2 simp3 957 . 2  |-  ( ( J  e.  Top  /\  A  e.  Fin  /\  X  =  Y )  ->  X  =  Y )
3 simpl1 958 . . . . 5  |-  ( ( ( J  e.  Top  /\  A  e.  Fin  /\  X  =  Y )  /\  x  e.  X
)  ->  J  e.  Top )
4 finlocfin.1 . . . . . 6  |-  X  = 
U. J
54topopn 16652 . . . . 5  |-  ( J  e.  Top  ->  X  e.  J )
63, 5syl 15 . . . 4  |-  ( ( ( J  e.  Top  /\  A  e.  Fin  /\  X  =  Y )  /\  x  e.  X
)  ->  X  e.  J )
7 simpr 447 . . . 4  |-  ( ( ( J  e.  Top  /\  A  e.  Fin  /\  X  =  Y )  /\  x  e.  X
)  ->  x  e.  X )
8 simpl2 959 . . . . 5  |-  ( ( ( J  e.  Top  /\  A  e.  Fin  /\  X  =  Y )  /\  x  e.  X
)  ->  A  e.  Fin )
9 ssrab2 3258 . . . . 5  |-  { s  e.  A  |  ( s  i^i  X )  =/=  (/) }  C_  A
10 ssfi 7083 . . . . 5  |-  ( ( A  e.  Fin  /\  { s  e.  A  | 
( s  i^i  X
)  =/=  (/) }  C_  A )  ->  { s  e.  A  |  ( s  i^i  X )  =/=  (/) }  e.  Fin )
118, 9, 10sylancl 643 . . . 4  |-  ( ( ( J  e.  Top  /\  A  e.  Fin  /\  X  =  Y )  /\  x  e.  X
)  ->  { s  e.  A  |  (
s  i^i  X )  =/=  (/) }  e.  Fin )
12 eleq2 2344 . . . . . 6  |-  ( n  =  X  ->  (
x  e.  n  <->  x  e.  X ) )
13 ineq2 3364 . . . . . . . . 9  |-  ( n  =  X  ->  (
s  i^i  n )  =  ( s  i^i 
X ) )
1413neeq1d 2459 . . . . . . . 8  |-  ( n  =  X  ->  (
( s  i^i  n
)  =/=  (/)  <->  ( s  i^i  X )  =/=  (/) ) )
1514rabbidv 2780 . . . . . . 7  |-  ( n  =  X  ->  { s  e.  A  |  ( s  i^i  n )  =/=  (/) }  =  {
s  e.  A  | 
( s  i^i  X
)  =/=  (/) } )
1615eleq1d 2349 . . . . . 6  |-  ( n  =  X  ->  ( { s  e.  A  |  ( s  i^i  n )  =/=  (/) }  e.  Fin 
<->  { s  e.  A  |  ( s  i^i 
X )  =/=  (/) }  e.  Fin ) )
1712, 16anbi12d 691 . . . . 5  |-  ( n  =  X  ->  (
( x  e.  n  /\  { s  e.  A  |  ( s  i^i  n )  =/=  (/) }  e.  Fin )  <->  ( x  e.  X  /\  { s  e.  A  |  ( s  i^i  X )  =/=  (/) }  e.  Fin ) ) )
1817rspcev 2884 . . . 4  |-  ( ( X  e.  J  /\  ( x  e.  X  /\  { s  e.  A  |  ( s  i^i 
X )  =/=  (/) }  e.  Fin ) )  ->  E. n  e.  J  ( x  e.  n  /\  { s  e.  A  |  ( s  i^i  n )  =/=  (/) }  e.  Fin ) )
196, 7, 11, 18syl12anc 1180 . . 3  |-  ( ( ( J  e.  Top  /\  A  e.  Fin  /\  X  =  Y )  /\  x  e.  X
)  ->  E. n  e.  J  ( x  e.  n  /\  { s  e.  A  |  ( s  i^i  n )  =/=  (/) }  e.  Fin ) )
2019ralrimiva 2626 . 2  |-  ( ( J  e.  Top  /\  A  e.  Fin  /\  X  =  Y )  ->  A. x  e.  X  E. n  e.  J  ( x  e.  n  /\  { s  e.  A  |  ( s  i^i  n )  =/=  (/) }  e.  Fin ) )
21 finlocfin.2 . . 3  |-  Y  = 
U. A
224, 21islocfin 26296 . 2  |-  ( A  e.  ( LocFin `  J
)  <->  ( J  e. 
Top  /\  X  =  Y  /\  A. x  e.  X  E. n  e.  J  ( x  e.  n  /\  { s  e.  A  |  ( s  i^i  n )  =/=  (/) }  e.  Fin ) ) )
231, 2, 20, 22syl3anbrc 1136 1  |-  ( ( J  e.  Top  /\  A  e.  Fin  /\  X  =  Y )  ->  A  e.  ( LocFin `  J )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   E.wrex 2544   {crab 2547    i^i cin 3151    C_ wss 3152   (/)c0 3455   U.cuni 3827   ` cfv 5255   Fincfn 6863   Topctop 16631   LocFinclocfin 26262
This theorem is referenced by:  locfincmp  26304
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-3or 935  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-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-er 6660  df-en 6864  df-fin 6867  df-top 16636  df-locfin 26266
  Copyright terms: Public domain W3C validator