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Theorem finlocfin 26417
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 958 . 2  |-  ( ( J  e.  Top  /\  A  e.  Fin  /\  X  =  Y )  ->  J  e.  Top )
2 simp3 960 . 2  |-  ( ( J  e.  Top  /\  A  e.  Fin  /\  X  =  Y )  ->  X  =  Y )
3 simpl1 961 . . . . 5  |-  ( ( ( J  e.  Top  /\  A  e.  Fin  /\  X  =  Y )  /\  x  e.  X
)  ->  J  e.  Top )
4 finlocfin.1 . . . . . 6  |-  X  = 
U. J
54topopn 17010 . . . . 5  |-  ( J  e.  Top  ->  X  e.  J )
63, 5syl 16 . . . 4  |-  ( ( ( J  e.  Top  /\  A  e.  Fin  /\  X  =  Y )  /\  x  e.  X
)  ->  X  e.  J )
7 simpr 449 . . . 4  |-  ( ( ( J  e.  Top  /\  A  e.  Fin  /\  X  =  Y )  /\  x  e.  X
)  ->  x  e.  X )
8 simpl2 962 . . . . 5  |-  ( ( ( J  e.  Top  /\  A  e.  Fin  /\  X  =  Y )  /\  x  e.  X
)  ->  A  e.  Fin )
9 ssrab2 3414 . . . . 5  |-  { s  e.  A  |  ( s  i^i  X )  =/=  (/) }  C_  A
10 ssfi 7358 . . . . 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 645 . . . 4  |-  ( ( ( J  e.  Top  /\  A  e.  Fin  /\  X  =  Y )  /\  x  e.  X
)  ->  { s  e.  A  |  (
s  i^i  X )  =/=  (/) }  e.  Fin )
12 eleq2 2503 . . . . . 6  |-  ( n  =  X  ->  (
x  e.  n  <->  x  e.  X ) )
13 ineq2 3522 . . . . . . . . 9  |-  ( n  =  X  ->  (
s  i^i  n )  =  ( s  i^i 
X ) )
1413neeq1d 2620 . . . . . . . 8  |-  ( n  =  X  ->  (
( s  i^i  n
)  =/=  (/)  <->  ( s  i^i  X )  =/=  (/) ) )
1514rabbidv 2954 . . . . . . 7  |-  ( n  =  X  ->  { s  e.  A  |  ( s  i^i  n )  =/=  (/) }  =  {
s  e.  A  | 
( s  i^i  X
)  =/=  (/) } )
1615eleq1d 2508 . . . . . 6  |-  ( n  =  X  ->  ( { s  e.  A  |  ( s  i^i  n )  =/=  (/) }  e.  Fin 
<->  { s  e.  A  |  ( s  i^i 
X )  =/=  (/) }  e.  Fin ) )
1712, 16anbi12d 693 . . . . 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 3058 . . . 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 1183 . . 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 2795 . 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 26414 . 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 1139 1  |-  ( ( J  e.  Top  /\  A  e.  Fin  /\  X  =  Y )  ->  A  e.  ( LocFin `  J )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1727    =/= wne 2605   A.wral 2711   E.wrex 2712   {crab 2715    i^i cin 3305    C_ wss 3306   (/)c0 3613   U.cuni 4039   ` cfv 5483   Fincfn 7138   Topctop 16989   LocFinclocfin 26380
This theorem is referenced by:  locfincmp  26422
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-rex 2717  df-rab 2720  df-v 2964  df-sbc 3168  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-pss 3322  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-tp 3846  df-op 3847  df-uni 4040  df-br 4238  df-opab 4292  df-mpt 4293  df-tr 4328  df-eprel 4523  df-id 4527  df-po 4532  df-so 4533  df-fr 4570  df-we 4572  df-ord 4613  df-on 4614  df-lim 4615  df-suc 4616  df-om 4875  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-f1 5488  df-fo 5489  df-f1o 5490  df-fv 5491  df-er 6934  df-en 7139  df-fin 7142  df-top 16994  df-locfin 26384
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