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Theorem locfincf 20545
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 19940 . . . . 5  |-  ( K  e.  (TopOn `  X
)  ->  K  e.  Top )
21ad2antrr 730 . . . 4  |-  ( ( ( K  e.  (TopOn `  X )  /\  J  C_  K )  /\  x  e.  ( LocFin `  J )
)  ->  K  e.  Top )
3 toponuni 19941 . . . . . 6  |-  ( K  e.  (TopOn `  X
)  ->  X  =  U. K )
43ad2antrr 730 . . . . 5  |-  ( ( ( K  e.  (TopOn `  X )  /\  J  C_  K )  /\  x  e.  ( LocFin `  J )
)  ->  X  =  U. K )
5 locfincf.1 . . . . . . 7  |-  X  = 
U. J
6 eqid 2422 . . . . . . 7  |-  U. x  =  U. x
75, 6locfinbas 20536 . . . . . 6  |-  ( x  e.  ( LocFin `  J
)  ->  X  =  U. x )
87adantl 467 . . . . 5  |-  ( ( ( K  e.  (TopOn `  X )  /\  J  C_  K )  /\  x  e.  ( LocFin `  J )
)  ->  X  =  U. x )
94, 8eqtr3d 2465 . . . 4  |-  ( ( ( K  e.  (TopOn `  X )  /\  J  C_  K )  /\  x  e.  ( LocFin `  J )
)  ->  U. K  = 
U. x )
104eleq2d 2492 . . . . . 6  |-  ( ( ( K  e.  (TopOn `  X )  /\  J  C_  K )  /\  x  e.  ( LocFin `  J )
)  ->  ( y  e.  X  <->  y  e.  U. K ) )
115locfinnei 20537 . . . . . . . 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 435 . . . . . . 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 3526 . . . . . . . 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 467 . . . . . . 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 662 . . . . . 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 238 . . . . 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 2834 . . . 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 2422 . . . . 5  |-  U. K  =  U. K
1918, 6islocfin 20531 . . . 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 1189 . . 3  |-  ( ( ( K  e.  (TopOn `  X )  /\  J  C_  K )  /\  x  e.  ( LocFin `  J )
)  ->  x  e.  ( LocFin `  K )
)
2120ex 435 . 2  |-  ( ( K  e.  (TopOn `  X )  /\  J  C_  K )  ->  (
x  e.  ( LocFin `  J )  ->  x  e.  ( LocFin `  K )
) )
2221ssrdv 3470 1  |-  ( ( K  e.  (TopOn `  X )  /\  J  C_  K )  ->  ( LocFin `
 J )  C_  ( LocFin `  K )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1872    =/= wne 2614   A.wral 2771   E.wrex 2772   {crab 2775    i^i cin 3435    C_ wss 3436   (/)c0 3761   U.cuni 4219   ` cfv 5601   Fincfn 7581   Topctop 19916  TopOnctopon 19917   LocFinclocfin 20518
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6598
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-rab 2780  df-v 3082  df-sbc 3300  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-op 4005  df-uni 4220  df-br 4424  df-opab 4483  df-mpt 4484  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-iota 5565  df-fun 5603  df-fv 5609  df-top 19920  df-topon 19922  df-locfin 20521
This theorem is referenced by: (None)
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