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Theorem locfincf 26340
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 16981 . . . . 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 16982 . . . . . 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 2435 . . . . . . 7  |-  U. x  =  U. x
75, 6locfinbas 26335 . . . . . 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 2469 . . . 4  |-  ( ( ( K  e.  (TopOn `  X )  /\  J  C_  K )  /\  x  e.  ( LocFin `  J )
)  ->  U. K  = 
U. x )
104eleq2d 2502 . . . . . 6  |-  ( ( ( K  e.  (TopOn `  X )  /\  J  C_  K )  /\  x  e.  ( LocFin `  J )
)  ->  ( y  e.  X  <->  y  e.  U. K ) )
115locfinnei 26336 . . . . . . . 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 3400 . . . . . . . 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 2780 . . . 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 2435 . . . . 5  |-  U. K  =  U. K
1918, 6islocfin 26330 . . . 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 3346 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 1652    e. wcel 1725    =/= wne 2598   A.wral 2697   E.wrex 2698   {crab 2701    i^i cin 3311    C_ wss 3312   (/)c0 3620   U.cuni 4007   ` cfv 5446   Fincfn 7101   Topctop 16948  TopOnctopon 16949   LocFinclocfin 26296
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fv 5454  df-top 16953  df-topon 16956  df-locfin 26300
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