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Theorem finlocfin 26277
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 957 . 2  |-  ( ( J  e.  Top  /\  A  e.  Fin  /\  X  =  Y )  ->  J  e.  Top )
2 simp3 959 . 2  |-  ( ( J  e.  Top  /\  A  e.  Fin  /\  X  =  Y )  ->  X  =  Y )
3 simpl1 960 . . . . 5  |-  ( ( ( J  e.  Top  /\  A  e.  Fin  /\  X  =  Y )  /\  x  e.  X
)  ->  J  e.  Top )
4 finlocfin.1 . . . . . 6  |-  X  = 
U. J
54topopn 16942 . . . . 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 448 . . . 4  |-  ( ( ( J  e.  Top  /\  A  e.  Fin  /\  X  =  Y )  /\  x  e.  X
)  ->  x  e.  X )
8 simpl2 961 . . . . 5  |-  ( ( ( J  e.  Top  /\  A  e.  Fin  /\  X  =  Y )  /\  x  e.  X
)  ->  A  e.  Fin )
9 ssrab2 3396 . . . . 5  |-  { s  e.  A  |  ( s  i^i  X )  =/=  (/) }  C_  A
10 ssfi 7296 . . . . 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 644 . . . 4  |-  ( ( ( J  e.  Top  /\  A  e.  Fin  /\  X  =  Y )  /\  x  e.  X
)  ->  { s  e.  A  |  (
s  i^i  X )  =/=  (/) }  e.  Fin )
12 eleq2 2473 . . . . . 6  |-  ( n  =  X  ->  (
x  e.  n  <->  x  e.  X ) )
13 ineq2 3504 . . . . . . . . 9  |-  ( n  =  X  ->  (
s  i^i  n )  =  ( s  i^i 
X ) )
1413neeq1d 2588 . . . . . . . 8  |-  ( n  =  X  ->  (
( s  i^i  n
)  =/=  (/)  <->  ( s  i^i  X )  =/=  (/) ) )
1514rabbidv 2916 . . . . . . 7  |-  ( n  =  X  ->  { s  e.  A  |  ( s  i^i  n )  =/=  (/) }  =  {
s  e.  A  | 
( s  i^i  X
)  =/=  (/) } )
1615eleq1d 2478 . . . . . 6  |-  ( n  =  X  ->  ( { s  e.  A  |  ( s  i^i  n )  =/=  (/) }  e.  Fin 
<->  { s  e.  A  |  ( s  i^i 
X )  =/=  (/) }  e.  Fin ) )
1712, 16anbi12d 692 . . . . 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 3020 . . . 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 1182 . . 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 2757 . 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 26274 . 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 1138 1  |-  ( ( J  e.  Top  /\  A  e.  Fin  /\  X  =  Y )  ->  A  e.  ( LocFin `  J )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2575   A.wral 2674   E.wrex 2675   {crab 2678    i^i cin 3287    C_ wss 3288   (/)c0 3596   U.cuni 3983   ` cfv 5421   Fincfn 7076   Topctop 16921   LocFinclocfin 26240
This theorem is referenced by:  locfincmp  26282
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-sbc 3130  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-er 6872  df-en 7077  df-fin 7080  df-top 16926  df-locfin 26244
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