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Theorem locfincf 26306
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 16664 . . . . 5  |-  ( K  e.  (TopOn `  X
)  ->  K  e.  Top )
21ad2antrr 706 . . . 4  |-  ( ( ( K  e.  (TopOn `  X )  /\  J  C_  K )  /\  x  e.  ( LocFin `  J )
)  ->  K  e.  Top )
3 toponuni 16665 . . . . . 6  |-  ( K  e.  (TopOn `  X
)  ->  X  =  U. K )
43ad2antrr 706 . . . . 5  |-  ( ( ( K  e.  (TopOn `  X )  /\  J  C_  K )  /\  x  e.  ( LocFin `  J )
)  ->  X  =  U. K )
5 locfincf.1 . . . . . . 7  |-  X  = 
U. J
6 eqid 2283 . . . . . . 7  |-  U. x  =  U. x
75, 6locfinbas 26301 . . . . . 6  |-  ( x  e.  ( LocFin `  J
)  ->  X  =  U. x )
87adantl 452 . . . . 5  |-  ( ( ( K  e.  (TopOn `  X )  /\  J  C_  K )  /\  x  e.  ( LocFin `  J )
)  ->  X  =  U. x )
94, 8eqtr3d 2317 . . . 4  |-  ( ( ( K  e.  (TopOn `  X )  /\  J  C_  K )  /\  x  e.  ( LocFin `  J )
)  ->  U. K  = 
U. x )
104eleq2d 2350 . . . . . 6  |-  ( ( ( K  e.  (TopOn `  X )  /\  J  C_  K )  /\  x  e.  ( LocFin `  J )
)  ->  ( y  e.  X  <->  y  e.  U. K ) )
115locfinnei 26302 . . . . . . . 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 423 . . . . . . 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 3238 . . . . . . . 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 452 . . . . . . 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 639 . . . . . 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 226 . . . . 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 2625 . . . 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 2283 . . . . 5  |-  U. K  =  U. K
1918, 6islocfin 26296 . . . 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 1136 . . 3  |-  ( ( ( K  e.  (TopOn `  X )  /\  J  C_  K )  /\  x  e.  ( LocFin `  J )
)  ->  x  e.  ( LocFin `  K )
)
2120ex 423 . 2  |-  ( ( K  e.  (TopOn `  X )  /\  J  C_  K )  ->  (
x  e.  ( LocFin `  J )  ->  x  e.  ( LocFin `  K )
) )
2221ssrdv 3185 1  |-  ( ( K  e.  (TopOn `  X )  /\  J  C_  K )  ->  ( LocFin `
 J )  C_  ( LocFin `  K )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = 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  TopOnctopon 16632   LocFinclocfin 26262
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-sbc 2992  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-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-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fv 5263  df-top 16636  df-topon 16639  df-locfin 26266
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