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Theorem inttop3 25516
Description: The intersection of a family of topologies is a topology. (Contributed by FL, 19-Sep-2011.)
Assertion
Ref Expression
inttop3  |-  ( ( J  =/=  (/)  /\  J  C_ 
Top )  ->  |^| J  e.  Top )

Proof of Theorem inttop3
Dummy variable  i is distinct from all other variables.
StepHypRef Expression
1 intiin 3956 . 2  |-  |^| J  =  |^|_ i  e.  J  i
2 dfss3 3170 . . 3  |-  ( J 
C_  Top  <->  A. i  e.  J  i  e.  Top )
3 inttop2 25515 . . 3  |-  ( ( J  =/=  (/)  /\  A. i  e.  J  i  e.  Top )  ->  |^|_ i  e.  J  i  e.  Top )
42, 3sylan2b 461 . 2  |-  ( ( J  =/=  (/)  /\  J  C_ 
Top )  ->  |^|_ i  e.  J  i  e.  Top )
51, 4syl5eqel 2367 1  |-  ( ( J  =/=  (/)  /\  J  C_ 
Top )  ->  |^| J  e.  Top )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    e. wcel 1684    =/= wne 2446   A.wral 2543    C_ wss 3152   (/)c0 3455   |^|cint 3862   |^|_ciin 3906   Topctop 16631
This theorem is referenced by:  inttop4  25517  prtoptop  25549
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-v 2790  df-dif 3155  df-in 3159  df-ss 3166  df-nul 3456  df-pw 3627  df-uni 3828  df-int 3863  df-iin 3908  df-top 16636
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