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Theorem fiinopn 19279
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 4024 . . . . . . 7  |-  ( A  e.  Fin  ->  ( A  e.  ~P J  <->  A 
C_  J ) )
2 sseq1 3530 . . . . . . . . . . . . . 14  |-  ( x  =  A  ->  (
x  C_  J  <->  A  C_  J
) )
3 neeq1 2748 . . . . . . . . . . . . . 14  |-  ( x  =  A  ->  (
x  =/=  (/)  <->  A  =/=  (/) ) )
4 eleq1 2539 . . . . . . . . . . . . . 14  |-  ( x  =  A  ->  (
x  e.  Fin  <->  A  e.  Fin ) )
52, 3, 43anbi123d 1299 . . . . . . . . . . . . 13  |-  ( x  =  A  ->  (
( x  C_  J  /\  x  =/=  (/)  /\  x  e.  Fin )  <->  ( A  C_  J  /\  A  =/=  (/)  /\  A  e.  Fin ) ) )
6 inteq 4291 . . . . . . . . . . . . . . 15  |-  ( x  =  A  ->  |^| x  =  |^| A )
76eleq1d 2536 . . . . . . . . . . . . . 14  |-  ( x  =  A  ->  ( |^| x  e.  J  <->  |^| A  e.  J ) )
87imbi2d 316 . . . . . . . . . . . . 13  |-  ( x  =  A  ->  (
( J  e.  Top  ->  |^| x  e.  J
)  <->  ( J  e. 
Top  ->  |^| A  e.  J
) ) )
95, 8imbi12d 320 . . . . . . . . . . . 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 istop2g 19274 . . . . . . . . . . . . . . 15  |-  ( J  e.  Top  ->  ( J  e.  Top  <->  ( A. x ( x  C_  J  ->  U. x  e.  J
)  /\  A. x
( ( x  C_  J  /\  x  =/=  (/)  /\  x  e.  Fin )  ->  |^| x  e.  J ) ) ) )
1110ibi 241 . . . . . . . . . . . . . 14  |-  ( J  e.  Top  ->  ( A. x ( x  C_  J  ->  U. x  e.  J
)  /\  A. x
( ( x  C_  J  /\  x  =/=  (/)  /\  x  e.  Fin )  ->  |^| x  e.  J ) ) )
12 sp 1808 . . . . . . . . . . . . . . 15  |-  ( A. x ( ( x 
C_  J  /\  x  =/=  (/)  /\  x  e. 
Fin )  ->  |^| x  e.  J )  ->  (
( x  C_  J  /\  x  =/=  (/)  /\  x  e.  Fin )  ->  |^| x  e.  J ) )
1312adantl 466 . . . . . . . . . . . . . 14  |-  ( ( 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 ) )
1411, 13syl 16 . . . . . . . . . . . . 13  |-  ( J  e.  Top  ->  (
( x  C_  J  /\  x  =/=  (/)  /\  x  e.  Fin )  ->  |^| x  e.  J ) )
1514com12 31 . . . . . . . . . . . 12  |-  ( ( x  C_  J  /\  x  =/=  (/)  /\  x  e. 
Fin )  ->  ( J  e.  Top  ->  |^| x  e.  J ) )
169, 15vtoclg 3176 . . . . . . . . . . 11  |-  ( A  e.  ~P J  -> 
( ( A  C_  J  /\  A  =/=  (/)  /\  A  e.  Fin )  ->  ( J  e.  Top  ->  |^| A  e.  J ) ) )
1716com12 31 . . . . . . . . . 10  |-  ( ( A  C_  J  /\  A  =/=  (/)  /\  A  e. 
Fin )  ->  ( A  e.  ~P J  ->  ( J  e.  Top  ->  |^| A  e.  J
) ) )
18173exp 1195 . . . . . . . . 9  |-  ( A 
C_  J  ->  ( A  =/=  (/)  ->  ( A  e.  Fin  ->  ( A  e.  ~P J  ->  ( J  e.  Top  ->  |^| A  e.  J ) ) ) ) )
1918com3r 79 . . . . . . . 8  |-  ( A  e.  Fin  ->  ( A  C_  J  ->  ( A  =/=  (/)  ->  ( A  e.  ~P J  ->  ( J  e.  Top  ->  |^| A  e.  J ) ) ) ) )
2019com4r 86 . . . . . . 7  |-  ( A  e.  ~P J  -> 
( A  e.  Fin  ->  ( A  C_  J  ->  ( A  =/=  (/)  ->  ( J  e.  Top  ->  |^| A  e.  J ) ) ) ) )
211, 20syl6bir 229 . . . . . 6  |-  ( A  e.  Fin  ->  ( A  C_  J  ->  ( A  e.  Fin  ->  ( A  C_  J  ->  ( A  =/=  (/)  ->  ( J  e.  Top  ->  |^| A  e.  J ) ) ) ) ) )
2221pm2.43a 49 . . . . 5  |-  ( A  e.  Fin  ->  ( A  C_  J  ->  ( A  C_  J  ->  ( A  =/=  (/)  ->  ( J  e.  Top  ->  |^| A  e.  J ) ) ) ) )
2322com4l 84 . . . 4  |-  ( A 
C_  J  ->  ( A  C_  J  ->  ( A  =/=  (/)  ->  ( A  e.  Fin  ->  ( J  e.  Top  ->  |^| A  e.  J ) ) ) ) )
2423pm2.43i 47 . . 3  |-  ( A 
C_  J  ->  ( A  =/=  (/)  ->  ( A  e.  Fin  ->  ( J  e.  Top  ->  |^| A  e.  J ) ) ) )
25243imp 1190 . 2  |-  ( ( A  C_  J  /\  A  =/=  (/)  /\  A  e. 
Fin )  ->  ( J  e.  Top  ->  |^| A  e.  J ) )
2625com12 31 1  |-  ( J  e.  Top  ->  (
( A  C_  J  /\  A  =/=  (/)  /\  A  e.  Fin )  ->  |^| A  e.  J ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973   A.wal 1377    = wceq 1379    e. wcel 1767    =/= wne 2662    C_ wss 3481   (/)c0 3790   ~Pcpw 4016   U.cuni 4251   |^|cint 4288   Fincfn 7528   Topctop 19263
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-recs 7054  df-rdg 7088  df-1o 7142  df-oadd 7146  df-er 7323  df-en 7529  df-fin 7532  df-top 19268
This theorem is referenced by:  iinopn  19280  hauscmplem  19774  1stcfb  19814  txtube  20009
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