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Theorem inttop4 25620
Description: The intersection of two topologies is a topology. (Contributed by FL, 19-Sep-2011.)
Assertion
Ref Expression
inttop4  |-  ( ( J  e.  Top  /\  K  e.  Top )  ->  ( J  i^i  K
)  e.  Top )

Proof of Theorem inttop4
StepHypRef Expression
1 prssg 3786 . . . 4  |-  ( ( J  e.  Top  /\  K  e.  Top )  ->  ( ( J  e. 
Top  /\  K  e.  Top )  <->  { J ,  K }  C_  Top ) )
21biimpd 198 . . 3  |-  ( ( J  e.  Top  /\  K  e.  Top )  ->  ( ( J  e. 
Top  /\  K  e.  Top )  ->  { J ,  K }  C_  Top ) )
3 intprg 3912 . . . . . . 7  |-  ( ( J  e.  Top  /\  K  e.  Top )  ->  |^| { J ,  K }  =  ( J  i^i  K ) )
43eqcomd 2301 . . . . . 6  |-  ( ( J  e.  Top  /\  K  e.  Top )  ->  ( J  i^i  K
)  =  |^| { J ,  K } )
54adantl 452 . . . . 5  |-  ( ( { J ,  K }  C_  Top  /\  ( J  e.  Top  /\  K  e.  Top ) )  -> 
( J  i^i  K
)  =  |^| { J ,  K } )
6 prnzg 3759 . . . . . . . 8  |-  ( J  e.  Top  ->  { J ,  K }  =/=  (/) )
7 inttop3 25619 . . . . . . . . 9  |-  ( ( { J ,  K }  =/=  (/)  /\  { J ,  K }  C_  Top )  ->  |^| { J ,  K }  e.  Top )
87ex 423 . . . . . . . 8  |-  ( { J ,  K }  =/=  (/)  ->  ( { J ,  K }  C_ 
Top  ->  |^| { J ,  K }  e.  Top ) )
96, 8syl 15 . . . . . . 7  |-  ( J  e.  Top  ->  ( { J ,  K }  C_ 
Top  ->  |^| { J ,  K }  e.  Top ) )
109adantr 451 . . . . . 6  |-  ( ( J  e.  Top  /\  K  e.  Top )  ->  ( { J ,  K }  C_  Top  ->  |^|
{ J ,  K }  e.  Top )
)
1110impcom 419 . . . . 5  |-  ( ( { J ,  K }  C_  Top  /\  ( J  e.  Top  /\  K  e.  Top ) )  ->  |^| { J ,  K }  e.  Top )
125, 11eqeltrd 2370 . . . 4  |-  ( ( { J ,  K }  C_  Top  /\  ( J  e.  Top  /\  K  e.  Top ) )  -> 
( J  i^i  K
)  e.  Top )
1312ex 423 . . 3  |-  ( { J ,  K }  C_ 
Top  ->  ( ( J  e.  Top  /\  K  e.  Top )  ->  ( J  i^i  K )  e. 
Top ) )
142, 13syli 33 . 2  |-  ( ( J  e.  Top  /\  K  e.  Top )  ->  ( ( J  e. 
Top  /\  K  e.  Top )  ->  ( J  i^i  K )  e. 
Top ) )
1514pm2.43i 43 1  |-  ( ( J  e.  Top  /\  K  e.  Top )  ->  ( J  i^i  K
)  e.  Top )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459    i^i cin 3164    C_ wss 3165   (/)c0 3468   {cpr 3654   |^|cint 3878   Topctop 16647
This theorem is referenced by:  intcont  25646
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-pw 3640  df-sn 3659  df-pr 3660  df-uni 3844  df-int 3879  df-iin 3924  df-top 16652
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