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Theorem fiinopn 13941
Description: The intersection of a nonempty finite family of open sets is open. (Contributed by FL, 20-Apr-2012.)
Assertion
Ref Expression
fiinopn  |-  ( J  e.  Top  ->  (
( A  C_  J  /\  A  =/=  (/)  /\  A  e.  Fin )  ->  |^| A  e.  J ) )

Proof of Theorem fiinopn
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 elpwg 3598 . . . . . . 7  |-  ( A  e.  Fin  ->  ( A  e.  ~P J  <->  A 
C_  J ) )
2 sseq1 3193 . . . . . . . . . . . . . 14  |-  ( x  =  A  ->  (
x  C_  J  <->  A  C_  J
) )
3 neeq1 2373 . . . . . . . . . . . . . 14  |-  ( x  =  A  ->  (
x  =/=  (/)  <->  A  =/=  (/) ) )
4 eleq1 2252 . . . . . . . . . . . . . 14  |-  ( x  =  A  ->  (
x  e.  Fin  <->  A  e.  Fin ) )
52, 3, 43anbi123d 1323 . . . . . . . . . . . . 13  |-  ( x  =  A  ->  (
( x  C_  J  /\  x  =/=  (/)  /\  x  e.  Fin )  <->  ( A  C_  J  /\  A  =/=  (/)  /\  A  e.  Fin ) ) )
6 inteq 3862 . . . . . . . . . . . . . . 15  |-  ( x  =  A  ->  |^| x  =  |^| A )
76eleq1d 2258 . . . . . . . . . . . . . 14  |-  ( x  =  A  ->  ( |^| x  e.  J  <->  |^| A  e.  J ) )
87imbi2d 230 . . . . . . . . . . . . 13  |-  ( x  =  A  ->  (
( J  e.  Top  ->  |^| x  e.  J
)  <->  ( J  e. 
Top  ->  |^| A  e.  J
) ) )
95, 8imbi12d 234 . . . . . . . . . . . 12  |-  ( x  =  A  ->  (
( ( x  C_  J  /\  x  =/=  (/)  /\  x  e.  Fin )  ->  ( J  e.  Top  ->  |^| x  e.  J ) )  <->  ( ( A  C_  J  /\  A  =/=  (/)  /\  A  e. 
Fin )  ->  ( J  e.  Top  ->  |^| A  e.  J ) ) ) )
10 sp 1522 . . . . . . . . . . . . . 14  |-  ( A. x ( ( x 
C_  J  /\  x  =/=  (/)  /\  x  e. 
Fin )  ->  |^| x  e.  J )  ->  (
( x  C_  J  /\  x  =/=  (/)  /\  x  e.  Fin )  ->  |^| x  e.  J ) )
1110adantl 277 . . . . . . . . . . . . 13  |-  ( ( A. x ( x 
C_  J  ->  U. x  e.  J )  /\  A. x ( ( x 
C_  J  /\  x  =/=  (/)  /\  x  e. 
Fin )  ->  |^| x  e.  J ) )  -> 
( ( x  C_  J  /\  x  =/=  (/)  /\  x  e.  Fin )  ->  |^| x  e.  J ) )
12 istopfin 13937 . . . . . . . . . . . . 13  |-  ( J  e.  Top  ->  ( A. x ( x  C_  J  ->  U. x  e.  J
)  /\  A. x
( ( x  C_  J  /\  x  =/=  (/)  /\  x  e.  Fin )  ->  |^| x  e.  J ) ) )
1311, 12syl11 31 . . . . . . . . . . . 12  |-  ( ( x  C_  J  /\  x  =/=  (/)  /\  x  e. 
Fin )  ->  ( J  e.  Top  ->  |^| x  e.  J ) )
149, 13vtoclg 2812 . . . . . . . . . . 11  |-  ( A  e.  ~P J  -> 
( ( A  C_  J  /\  A  =/=  (/)  /\  A  e.  Fin )  ->  ( J  e.  Top  ->  |^| A  e.  J ) ) )
1514com12 30 . . . . . . . . . 10  |-  ( ( A  C_  J  /\  A  =/=  (/)  /\  A  e. 
Fin )  ->  ( A  e.  ~P J  ->  ( J  e.  Top  ->  |^| A  e.  J
) ) )
16153exp 1204 . . . . . . . . 9  |-  ( A 
C_  J  ->  ( A  =/=  (/)  ->  ( A  e.  Fin  ->  ( A  e.  ~P J  ->  ( J  e.  Top  ->  |^| A  e.  J ) ) ) ) )
1716com3r 79 . . . . . . . 8  |-  ( A  e.  Fin  ->  ( A  C_  J  ->  ( A  =/=  (/)  ->  ( A  e.  ~P J  ->  ( J  e.  Top  ->  |^| A  e.  J ) ) ) ) )
1817com4r 86 . . . . . . 7  |-  ( A  e.  ~P J  -> 
( A  e.  Fin  ->  ( A  C_  J  ->  ( A  =/=  (/)  ->  ( J  e.  Top  ->  |^| A  e.  J ) ) ) ) )
191, 18biimtrrdi 164 . . . . . 6  |-  ( A  e.  Fin  ->  ( A  C_  J  ->  ( A  e.  Fin  ->  ( A  C_  J  ->  ( A  =/=  (/)  ->  ( J  e.  Top  ->  |^| A  e.  J ) ) ) ) ) )
2019pm2.43a 51 . . . . 5  |-  ( A  e.  Fin  ->  ( A  C_  J  ->  ( A  C_  J  ->  ( A  =/=  (/)  ->  ( J  e.  Top  ->  |^| A  e.  J ) ) ) ) )
2120com4l 84 . . . 4  |-  ( A 
C_  J  ->  ( A  C_  J  ->  ( A  =/=  (/)  ->  ( A  e.  Fin  ->  ( J  e.  Top  ->  |^| A  e.  J ) ) ) ) )
2221pm2.43i 49 . . 3  |-  ( A 
C_  J  ->  ( A  =/=  (/)  ->  ( A  e.  Fin  ->  ( J  e.  Top  ->  |^| A  e.  J ) ) ) )
23223imp 1195 . 2  |-  ( ( A  C_  J  /\  A  =/=  (/)  /\  A  e. 
Fin )  ->  ( J  e.  Top  ->  |^| A  e.  J ) )
2423com12 30 1  |-  ( J  e.  Top  ->  (
( A  C_  J  /\  A  =/=  (/)  /\  A  e.  Fin )  ->  |^| A  e.  J ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    /\ w3a 980   A.wal 1362    = wceq 1364    e. wcel 2160    =/= wne 2360    C_ wss 3144   (/)c0 3437   ~Pcpw 3590   U.cuni 3824   |^|cint 3859   Fincfn 6761   Topctop 13934
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-nul 4144  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-iinf 4602
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-v 2754  df-sbc 2978  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-br 4019  df-opab 4080  df-id 4308  df-suc 4386  df-iom 4605  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5234  df-fn 5235  df-f 5236  df-f1 5237  df-fo 5238  df-f1o 5239  df-fv 5240  df-er 6554  df-en 6762  df-fin 6764  df-top 13935
This theorem is referenced by: (None)
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