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Theorem finlocfin 25806
Description: A finite cover of a topological space is a locally finite cover. (Contributed by Jeff Hankins, 21-Jan-2010.)
Hypotheses
Ref Expression
finlocfin.1  |-  X  = 
U. J
finlocfin.2  |-  Y  = 
U. A
Assertion
Ref Expression
finlocfin  |-  ( ( J  e.  Top  /\  A  e.  Fin  /\  X  =  Y )  ->  A  e.  ( LocFin `  J )
)

Proof of Theorem finlocfin
Dummy variables  n  s  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp1 956 . 2  |-  ( ( J  e.  Top  /\  A  e.  Fin  /\  X  =  Y )  ->  J  e.  Top )
2 simp3 958 . 2  |-  ( ( J  e.  Top  /\  A  e.  Fin  /\  X  =  Y )  ->  X  =  Y )
3 simpl1 959 . . . . 5  |-  ( ( ( J  e.  Top  /\  A  e.  Fin  /\  X  =  Y )  /\  x  e.  X
)  ->  J  e.  Top )
4 finlocfin.1 . . . . . 6  |-  X  = 
U. J
54topopn 16869 . . . . 5  |-  ( J  e.  Top  ->  X  e.  J )
63, 5syl 15 . . . 4  |-  ( ( ( J  e.  Top  /\  A  e.  Fin  /\  X  =  Y )  /\  x  e.  X
)  ->  X  e.  J )
7 simpr 447 . . . 4  |-  ( ( ( J  e.  Top  /\  A  e.  Fin  /\  X  =  Y )  /\  x  e.  X
)  ->  x  e.  X )
8 simpl2 960 . . . . 5  |-  ( ( ( J  e.  Top  /\  A  e.  Fin  /\  X  =  Y )  /\  x  e.  X
)  ->  A  e.  Fin )
9 ssrab2 3344 . . . . 5  |-  { s  e.  A  |  ( s  i^i  X )  =/=  (/) }  C_  A
10 ssfi 7226 . . . . 5  |-  ( ( A  e.  Fin  /\  { s  e.  A  | 
( s  i^i  X
)  =/=  (/) }  C_  A )  ->  { s  e.  A  |  ( s  i^i  X )  =/=  (/) }  e.  Fin )
118, 9, 10sylancl 643 . . . 4  |-  ( ( ( J  e.  Top  /\  A  e.  Fin  /\  X  =  Y )  /\  x  e.  X
)  ->  { s  e.  A  |  (
s  i^i  X )  =/=  (/) }  e.  Fin )
12 eleq2 2427 . . . . . 6  |-  ( n  =  X  ->  (
x  e.  n  <->  x  e.  X ) )
13 ineq2 3452 . . . . . . . . 9  |-  ( n  =  X  ->  (
s  i^i  n )  =  ( s  i^i 
X ) )
1413neeq1d 2542 . . . . . . . 8  |-  ( n  =  X  ->  (
( s  i^i  n
)  =/=  (/)  <->  ( s  i^i  X )  =/=  (/) ) )
1514rabbidv 2865 . . . . . . 7  |-  ( n  =  X  ->  { s  e.  A  |  ( s  i^i  n )  =/=  (/) }  =  {
s  e.  A  | 
( s  i^i  X
)  =/=  (/) } )
1615eleq1d 2432 . . . . . 6  |-  ( n  =  X  ->  ( { s  e.  A  |  ( s  i^i  n )  =/=  (/) }  e.  Fin 
<->  { s  e.  A  |  ( s  i^i 
X )  =/=  (/) }  e.  Fin ) )
1712, 16anbi12d 691 . . . . 5  |-  ( n  =  X  ->  (
( x  e.  n  /\  { s  e.  A  |  ( s  i^i  n )  =/=  (/) }  e.  Fin )  <->  ( x  e.  X  /\  { s  e.  A  |  ( s  i^i  X )  =/=  (/) }  e.  Fin ) ) )
1817rspcev 2969 . . . 4  |-  ( ( X  e.  J  /\  ( x  e.  X  /\  { s  e.  A  |  ( s  i^i 
X )  =/=  (/) }  e.  Fin ) )  ->  E. n  e.  J  ( x  e.  n  /\  { s  e.  A  |  ( s  i^i  n )  =/=  (/) }  e.  Fin ) )
196, 7, 11, 18syl12anc 1181 . . 3  |-  ( ( ( J  e.  Top  /\  A  e.  Fin  /\  X  =  Y )  /\  x  e.  X
)  ->  E. n  e.  J  ( x  e.  n  /\  { s  e.  A  |  ( s  i^i  n )  =/=  (/) }  e.  Fin ) )
2019ralrimiva 2711 . 2  |-  ( ( J  e.  Top  /\  A  e.  Fin  /\  X  =  Y )  ->  A. x  e.  X  E. n  e.  J  ( x  e.  n  /\  { s  e.  A  |  ( s  i^i  n )  =/=  (/) }  e.  Fin ) )
21 finlocfin.2 . . 3  |-  Y  = 
U. A
224, 21islocfin 25803 . 2  |-  ( A  e.  ( LocFin `  J
)  <->  ( J  e. 
Top  /\  X  =  Y  /\  A. x  e.  X  E. n  e.  J  ( x  e.  n  /\  { s  e.  A  |  ( s  i^i  n )  =/=  (/) }  e.  Fin ) ) )
231, 2, 20, 22syl3anbrc 1137 1  |-  ( ( J  e.  Top  /\  A  e.  Fin  /\  X  =  Y )  ->  A  e.  ( LocFin `  J )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 935    = wceq 1647    e. wcel 1715    =/= wne 2529   A.wral 2628   E.wrex 2629   {crab 2632    i^i cin 3237    C_ wss 3238   (/)c0 3543   U.cuni 3929   ` cfv 5358   Fincfn 7006   Topctop 16848   LocFinclocfin 25769
This theorem is referenced by:  locfincmp  25811
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-rab 2637  df-v 2875  df-sbc 3078  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-pss 3254  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-tp 3737  df-op 3738  df-uni 3930  df-br 4126  df-opab 4180  df-mpt 4181  df-tr 4216  df-eprel 4408  df-id 4412  df-po 4417  df-so 4418  df-fr 4455  df-we 4457  df-ord 4498  df-on 4499  df-lim 4500  df-suc 4501  df-om 4760  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-er 6802  df-en 7007  df-fin 7010  df-top 16853  df-locfin 25773
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