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Theorem fiinopn 12413
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 3551 . . . . . . 7  |-  ( A  e.  Fin  ->  ( A  e.  ~P J  <->  A 
C_  J ) )
2 sseq1 3151 . . . . . . . . . . . . . 14  |-  ( x  =  A  ->  (
x  C_  J  <->  A  C_  J
) )
3 neeq1 2340 . . . . . . . . . . . . . 14  |-  ( x  =  A  ->  (
x  =/=  (/)  <->  A  =/=  (/) ) )
4 eleq1 2220 . . . . . . . . . . . . . 14  |-  ( x  =  A  ->  (
x  e.  Fin  <->  A  e.  Fin ) )
52, 3, 43anbi123d 1294 . . . . . . . . . . . . 13  |-  ( x  =  A  ->  (
( x  C_  J  /\  x  =/=  (/)  /\  x  e.  Fin )  <->  ( A  C_  J  /\  A  =/=  (/)  /\  A  e.  Fin ) ) )
6 inteq 3810 . . . . . . . . . . . . . . 15  |-  ( x  =  A  ->  |^| x  =  |^| A )
76eleq1d 2226 . . . . . . . . . . . . . 14  |-  ( x  =  A  ->  ( |^| x  e.  J  <->  |^| A  e.  J ) )
87imbi2d 229 . . . . . . . . . . . . 13  |-  ( x  =  A  ->  (
( J  e.  Top  ->  |^| x  e.  J
)  <->  ( J  e. 
Top  ->  |^| A  e.  J
) ) )
95, 8imbi12d 233 . . . . . . . . . . . 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 1491 . . . . . . . . . . . . . 14  |-  ( A. x ( ( x 
C_  J  /\  x  =/=  (/)  /\  x  e. 
Fin )  ->  |^| x  e.  J )  ->  (
( x  C_  J  /\  x  =/=  (/)  /\  x  e.  Fin )  ->  |^| x  e.  J ) )
1110adantl 275 . . . . . . . . . . . . 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 12409 . . . . . . . . . . . . 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 2772 . . . . . . . . . . 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 1184 . . . . . . . . 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, 18syl6bir 163 . . . . . 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 1176 . 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 103    /\ w3a 963   A.wal 1333    = wceq 1335    e. wcel 2128    =/= wne 2327    C_ wss 3102   (/)c0 3394   ~Pcpw 3543   U.cuni 3772   |^|cint 3807   Fincfn 6685   Topctop 12406
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-nul 4090  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-iinf 4547
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-v 2714  df-sbc 2938  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-int 3808  df-br 3966  df-opab 4026  df-id 4253  df-suc 4331  df-iom 4550  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-rn 4597  df-res 4598  df-ima 4599  df-iota 5135  df-fun 5172  df-fn 5173  df-f 5174  df-f1 5175  df-fo 5176  df-f1o 5177  df-fv 5178  df-er 6480  df-en 6686  df-fin 6688  df-top 12407
This theorem is referenced by: (None)
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