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Theorem fiinopn 12796
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 3574 . . . . . . 7  |-  ( A  e.  Fin  ->  ( A  e.  ~P J  <->  A 
C_  J ) )
2 sseq1 3170 . . . . . . . . . . . . . 14  |-  ( x  =  A  ->  (
x  C_  J  <->  A  C_  J
) )
3 neeq1 2353 . . . . . . . . . . . . . 14  |-  ( x  =  A  ->  (
x  =/=  (/)  <->  A  =/=  (/) ) )
4 eleq1 2233 . . . . . . . . . . . . . 14  |-  ( x  =  A  ->  (
x  e.  Fin  <->  A  e.  Fin ) )
52, 3, 43anbi123d 1307 . . . . . . . . . . . . 13  |-  ( x  =  A  ->  (
( x  C_  J  /\  x  =/=  (/)  /\  x  e.  Fin )  <->  ( A  C_  J  /\  A  =/=  (/)  /\  A  e.  Fin ) ) )
6 inteq 3834 . . . . . . . . . . . . . . 15  |-  ( x  =  A  ->  |^| x  =  |^| A )
76eleq1d 2239 . . . . . . . . . . . . . 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 1504 . . . . . . . . . . . . . 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 12792 . . . . . . . . . . . . 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 2790 . . . . . . . . . . 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 1197 . . . . . . . . 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 1188 . 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 973   A.wal 1346    = wceq 1348    e. wcel 2141    =/= wne 2340    C_ wss 3121   (/)c0 3414   ~Pcpw 3566   U.cuni 3796   |^|cint 3831   Fincfn 6718   Topctop 12789
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-iinf 4572
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-br 3990  df-opab 4051  df-id 4278  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-er 6513  df-en 6719  df-fin 6721  df-top 12790
This theorem is referenced by: (None)
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