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Theorem locfincf 26409
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 16680 . . . . 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 16681 . . . . . 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 2296 . . . . . . 7  |-  U. x  =  U. x
75, 6locfinbas 26404 . . . . . 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 2330 . . . 4  |-  ( ( ( K  e.  (TopOn `  X )  /\  J  C_  K )  /\  x  e.  ( LocFin `  J )
)  ->  U. K  = 
U. x )
104eleq2d 2363 . . . . . 6  |-  ( ( ( K  e.  (TopOn `  X )  /\  J  C_  K )  /\  x  e.  ( LocFin `  J )
)  ->  ( y  e.  X  <->  y  e.  U. K ) )
115locfinnei 26405 . . . . . . . 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 3251 . . . . . . . 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 2638 . . . 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 2296 . . . . 5  |-  U. K  =  U. K
1918, 6islocfin 26399 . . . 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 3198 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 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   E.wrex 2557   {crab 2560    i^i cin 3164    C_ wss 3165   (/)c0 3468   U.cuni 3843   ` cfv 5271   Fincfn 6879   Topctop 16647  TopOnctopon 16648   LocFinclocfin 26365
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fv 5279  df-top 16652  df-topon 16655  df-locfin 26369
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