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Theorem istopfin 13883
Description: Express the predicate " J is a topology" using nonempty finite intersections instead of binary intersections as in istopg 13882. It is not clear we can prove the converse without adding additional conditions. (Contributed by NM, 19-Jul-2006.) (Revised by Jim Kingdon, 14-Jan-2023.)
Assertion
Ref Expression
istopfin  |-  ( J  e.  Top  ->  ( A. x ( x  C_  J  ->  U. x  e.  J
)  /\  A. x
( ( x  C_  J  /\  x  =/=  (/)  /\  x  e.  Fin )  ->  |^| x  e.  J ) ) )
Distinct variable group:    x, J

Proof of Theorem istopfin
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 istopg 13882 . . 3  |-  ( J  e.  Top  ->  ( J  e.  Top  <->  ( A. x ( x  C_  J  ->  U. x  e.  J
)  /\  A. x  e.  J  A. y  e.  J  ( x  i^i  y )  e.  J
) ) )
21ibi 176 . 2  |-  ( J  e.  Top  ->  ( A. x ( x  C_  J  ->  U. x  e.  J
)  /\  A. x  e.  J  A. y  e.  J  ( x  i^i  y )  e.  J
) )
3 fiintim 6945 . . 3  |-  ( A. x  e.  J  A. y  e.  J  (
x  i^i  y )  e.  J  ->  A. x
( ( x  C_  J  /\  x  =/=  (/)  /\  x  e.  Fin )  ->  |^| x  e.  J ) )
43anim2i 342 . 2  |-  ( ( A. x ( x 
C_  J  ->  U. x  e.  J )  /\  A. x  e.  J  A. y  e.  J  (
x  i^i  y )  e.  J )  ->  ( A. x ( x  C_  J  ->  U. x  e.  J
)  /\  A. x
( ( x  C_  J  /\  x  =/=  (/)  /\  x  e.  Fin )  ->  |^| x  e.  J ) ) )
52, 4syl 14 1  |-  ( J  e.  Top  ->  ( A. x ( x  C_  J  ->  U. x  e.  J
)  /\  A. x
( ( x  C_  J  /\  x  =/=  (/)  /\  x  e.  Fin )  ->  |^| x  e.  J ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    /\ w3a 979   A.wal 1361    e. wcel 2159    =/= wne 2359   A.wral 2467    i^i cin 3142    C_ wss 3143   (/)c0 3436   U.cuni 3823   |^|cint 3858   Fincfn 6757   Topctop 13880
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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2161  ax-14 2162  ax-ext 2170  ax-sep 4135  ax-nul 4143  ax-pow 4188  ax-pr 4223  ax-un 4447  ax-iinf 4601
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2040  df-mo 2041  df-clab 2175  df-cleq 2181  df-clel 2184  df-nfc 2320  df-ne 2360  df-ral 2472  df-rex 2473  df-v 2753  df-sbc 2977  df-dif 3145  df-un 3147  df-in 3149  df-ss 3156  df-nul 3437  df-pw 3591  df-sn 3612  df-pr 3613  df-op 3615  df-uni 3824  df-int 3859  df-br 4018  df-opab 4079  df-id 4307  df-suc 4385  df-iom 4604  df-xp 4646  df-rel 4647  df-cnv 4648  df-co 4649  df-dm 4650  df-rn 4651  df-res 4652  df-ima 4653  df-iota 5192  df-fun 5232  df-fn 5233  df-f 5234  df-f1 5235  df-fo 5236  df-f1o 5237  df-fv 5238  df-er 6552  df-en 6758  df-fin 6760  df-top 13881
This theorem is referenced by:  fiinopn  13887
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