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Theorem locfincf 26079
Description: A locally finite cover in a coarser topology is locally finite in a finer topology. (Contributed by Jeff Hankins, 22-Jan-2010.) (Proof shortened by Mario Carneiro, 11-Sep-2015.)
Hypothesis
Ref Expression
locfincf.1  |-  X  = 
U. J
Assertion
Ref Expression
locfincf  |-  ( ( K  e.  (TopOn `  X )  /\  J  C_  K )  ->  ( LocFin `
 J )  C_  ( LocFin `  K )
)

Proof of Theorem locfincf
Dummy variables  n  s  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 topontop 16916 . . . . 5  |-  ( K  e.  (TopOn `  X
)  ->  K  e.  Top )
21ad2antrr 707 . . . 4  |-  ( ( ( K  e.  (TopOn `  X )  /\  J  C_  K )  /\  x  e.  ( LocFin `  J )
)  ->  K  e.  Top )
3 toponuni 16917 . . . . . 6  |-  ( K  e.  (TopOn `  X
)  ->  X  =  U. K )
43ad2antrr 707 . . . . 5  |-  ( ( ( K  e.  (TopOn `  X )  /\  J  C_  K )  /\  x  e.  ( LocFin `  J )
)  ->  X  =  U. K )
5 locfincf.1 . . . . . . 7  |-  X  = 
U. J
6 eqid 2389 . . . . . . 7  |-  U. x  =  U. x
75, 6locfinbas 26074 . . . . . 6  |-  ( x  e.  ( LocFin `  J
)  ->  X  =  U. x )
87adantl 453 . . . . 5  |-  ( ( ( K  e.  (TopOn `  X )  /\  J  C_  K )  /\  x  e.  ( LocFin `  J )
)  ->  X  =  U. x )
94, 8eqtr3d 2423 . . . 4  |-  ( ( ( K  e.  (TopOn `  X )  /\  J  C_  K )  /\  x  e.  ( LocFin `  J )
)  ->  U. K  = 
U. x )
104eleq2d 2456 . . . . . 6  |-  ( ( ( K  e.  (TopOn `  X )  /\  J  C_  K )  /\  x  e.  ( LocFin `  J )
)  ->  ( y  e.  X  <->  y  e.  U. K ) )
115locfinnei 26075 . . . . . . . 8  |-  ( ( x  e.  ( LocFin `  J )  /\  y  e.  X )  ->  E. n  e.  J  ( y  e.  n  /\  { s  e.  x  |  ( s  i^i  n )  =/=  (/) }  e.  Fin ) )
1211ex 424 . . . . . . 7  |-  ( x  e.  ( LocFin `  J
)  ->  ( y  e.  X  ->  E. n  e.  J  ( y  e.  n  /\  { s  e.  x  |  ( s  i^i  n )  =/=  (/) }  e.  Fin ) ) )
13 ssrexv 3353 . . . . . . . 8  |-  ( J 
C_  K  ->  ( E. n  e.  J  ( y  e.  n  /\  { s  e.  x  |  ( s  i^i  n )  =/=  (/) }  e.  Fin )  ->  E. n  e.  K  ( y  e.  n  /\  { s  e.  x  |  ( s  i^i  n )  =/=  (/) }  e.  Fin ) ) )
1413adantl 453 . . . . . . 7  |-  ( ( K  e.  (TopOn `  X )  /\  J  C_  K )  ->  ( E. n  e.  J  ( y  e.  n  /\  { s  e.  x  |  ( s  i^i  n )  =/=  (/) }  e.  Fin )  ->  E. n  e.  K  ( y  e.  n  /\  { s  e.  x  |  ( s  i^i  n )  =/=  (/) }  e.  Fin ) ) )
1512, 14sylan9r 640 . . . . . 6  |-  ( ( ( K  e.  (TopOn `  X )  /\  J  C_  K )  /\  x  e.  ( LocFin `  J )
)  ->  ( y  e.  X  ->  E. n  e.  K  ( y  e.  n  /\  { s  e.  x  |  ( s  i^i  n )  =/=  (/) }  e.  Fin ) ) )
1610, 15sylbird 227 . . . . 5  |-  ( ( ( K  e.  (TopOn `  X )  /\  J  C_  K )  /\  x  e.  ( LocFin `  J )
)  ->  ( y  e.  U. K  ->  E. n  e.  K  ( y  e.  n  /\  { s  e.  x  |  ( s  i^i  n )  =/=  (/) }  e.  Fin ) ) )
1716ralrimiv 2733 . . . 4  |-  ( ( ( K  e.  (TopOn `  X )  /\  J  C_  K )  /\  x  e.  ( LocFin `  J )
)  ->  A. y  e.  U. K E. n  e.  K  ( y  e.  n  /\  { s  e.  x  |  ( s  i^i  n )  =/=  (/) }  e.  Fin ) )
18 eqid 2389 . . . . 5  |-  U. K  =  U. K
1918, 6islocfin 26069 . . . 4  |-  ( x  e.  ( LocFin `  K
)  <->  ( K  e. 
Top  /\  U. K  = 
U. x  /\  A. y  e.  U. K E. n  e.  K  (
y  e.  n  /\  { s  e.  x  |  ( s  i^i  n
)  =/=  (/) }  e.  Fin ) ) )
202, 9, 17, 19syl3anbrc 1138 . . 3  |-  ( ( ( K  e.  (TopOn `  X )  /\  J  C_  K )  /\  x  e.  ( LocFin `  J )
)  ->  x  e.  ( LocFin `  K )
)
2120ex 424 . 2  |-  ( ( K  e.  (TopOn `  X )  /\  J  C_  K )  ->  (
x  e.  ( LocFin `  J )  ->  x  e.  ( LocFin `  K )
) )
2221ssrdv 3299 1  |-  ( ( K  e.  (TopOn `  X )  /\  J  C_  K )  ->  ( LocFin `
 J )  C_  ( LocFin `  K )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717    =/= wne 2552   A.wral 2651   E.wrex 2652   {crab 2655    i^i cin 3264    C_ wss 3265   (/)c0 3573   U.cuni 3959   ` cfv 5396   Fincfn 7047   Topctop 16883  TopOnctopon 16884   LocFinclocfin 26035
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-rab 2660  df-v 2903  df-sbc 3107  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fv 5404  df-top 16888  df-topon 16891  df-locfin 26039
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