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Theorem fiinopn 16703
Description: The intersection of a non-empty 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 3666 . . . . . . 7  |-  ( A  e.  Fin  ->  ( A  e.  ~P J  <->  A 
C_  J ) )
2 sseq1 3233 . . . . . . . . . . . . . 14  |-  ( x  =  A  ->  (
x  C_  J  <->  A  C_  J
) )
3 neeq1 2487 . . . . . . . . . . . . . 14  |-  ( x  =  A  ->  (
x  =/=  (/)  <->  A  =/=  (/) ) )
4 eleq1 2376 . . . . . . . . . . . . . 14  |-  ( x  =  A  ->  (
x  e.  Fin  <->  A  e.  Fin ) )
52, 3, 43anbi123d 1252 . . . . . . . . . . . . 13  |-  ( x  =  A  ->  (
( x  C_  J  /\  x  =/=  (/)  /\  x  e.  Fin )  <->  ( A  C_  J  /\  A  =/=  (/)  /\  A  e.  Fin ) ) )
6 inteq 3902 . . . . . . . . . . . . . . 15  |-  ( x  =  A  ->  |^| x  =  |^| A )
76eleq1d 2382 . . . . . . . . . . . . . 14  |-  ( x  =  A  ->  ( |^| x  e.  J  <->  |^| A  e.  J ) )
87imbi2d 307 . . . . . . . . . . . . 13  |-  ( x  =  A  ->  (
( J  e.  Top  ->  |^| x  e.  J
)  <->  ( J  e. 
Top  ->  |^| A  e.  J
) ) )
95, 8imbi12d 311 . . . . . . . . . . . 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 16698 . . . . . . . . . . . . . . 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 232 . . . . . . . . . . . . . 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 1733 . . . . . . . . . . . . . . 15  |-  ( A. x ( ( x 
C_  J  /\  x  =/=  (/)  /\  x  e. 
Fin )  ->  |^| x  e.  J )  ->  (
( x  C_  J  /\  x  =/=  (/)  /\  x  e.  Fin )  ->  |^| x  e.  J ) )
1312adantl 452 . . . . . . . . . . . . . 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 15 . . . . . . . . . . . . 13  |-  ( J  e.  Top  ->  (
( x  C_  J  /\  x  =/=  (/)  /\  x  e.  Fin )  ->  |^| x  e.  J ) )
1514com12 27 . . . . . . . . . . . 12  |-  ( ( x  C_  J  /\  x  =/=  (/)  /\  x  e. 
Fin )  ->  ( J  e.  Top  ->  |^| x  e.  J ) )
169, 15vtoclg 2877 . . . . . . . . . . 11  |-  ( A  e.  ~P J  -> 
( ( A  C_  J  /\  A  =/=  (/)  /\  A  e.  Fin )  ->  ( J  e.  Top  ->  |^| A  e.  J ) ) )
1716com12 27 . . . . . . . . . 10  |-  ( ( A  C_  J  /\  A  =/=  (/)  /\  A  e. 
Fin )  ->  ( A  e.  ~P J  ->  ( J  e.  Top  ->  |^| A  e.  J
) ) )
18173exp 1150 . . . . . . . . 9  |-  ( A 
C_  J  ->  ( A  =/=  (/)  ->  ( A  e.  Fin  ->  ( A  e.  ~P J  ->  ( J  e.  Top  ->  |^| A  e.  J ) ) ) ) )
1918com3r 73 . . . . . . . 8  |-  ( A  e.  Fin  ->  ( A  C_  J  ->  ( A  =/=  (/)  ->  ( A  e.  ~P J  ->  ( J  e.  Top  ->  |^| A  e.  J ) ) ) ) )
2019com4r 80 . . . . . . 7  |-  ( A  e.  ~P J  -> 
( A  e.  Fin  ->  ( A  C_  J  ->  ( A  =/=  (/)  ->  ( J  e.  Top  ->  |^| A  e.  J ) ) ) ) )
211, 20syl6bir 220 . . . . . 6  |-  ( A  e.  Fin  ->  ( A  C_  J  ->  ( A  e.  Fin  ->  ( A  C_  J  ->  ( A  =/=  (/)  ->  ( J  e.  Top  ->  |^| A  e.  J ) ) ) ) ) )
2221pm2.43a 45 . . . . 5  |-  ( A  e.  Fin  ->  ( A  C_  J  ->  ( A  C_  J  ->  ( A  =/=  (/)  ->  ( J  e.  Top  ->  |^| A  e.  J ) ) ) ) )
2322com4l 78 . . . 4  |-  ( A 
C_  J  ->  ( A  C_  J  ->  ( A  =/=  (/)  ->  ( A  e.  Fin  ->  ( J  e.  Top  ->  |^| A  e.  J ) ) ) ) )
2423pm2.43i 43 . . 3  |-  ( A 
C_  J  ->  ( A  =/=  (/)  ->  ( A  e.  Fin  ->  ( J  e.  Top  ->  |^| A  e.  J ) ) ) )
25243imp 1145 . 2  |-  ( ( A  C_  J  /\  A  =/=  (/)  /\  A  e. 
Fin )  ->  ( J  e.  Top  ->  |^| A  e.  J ) )
2625com12 27 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 358    /\ w3a 934   A.wal 1531    = wceq 1633    e. wcel 1701    =/= wne 2479    C_ wss 3186   (/)c0 3489   ~Pcpw 3659   U.cuni 3864   |^|cint 3899   Fincfn 6906   Topctop 16687
This theorem is referenced by:  iinopn  16704  hauscmplem  17189  1stcfb  17227  txtube  17390
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-reu 2584  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-pss 3202  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-tp 3682  df-op 3683  df-uni 3865  df-int 3900  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-tr 4151  df-eprel 4342  df-id 4346  df-po 4351  df-so 4352  df-fr 4389  df-we 4391  df-ord 4432  df-on 4433  df-lim 4434  df-suc 4435  df-om 4694  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-recs 6430  df-rdg 6465  df-1o 6521  df-oadd 6525  df-er 6702  df-en 6907  df-fin 6910  df-top 16692
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