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Theorem riinopn 16986
Description: A finite indexed relative intersection of open sets is open. (Contributed by Mario Carneiro, 22-Aug-2015.)
Hypothesis
Ref Expression
1open.1  |-  X  = 
U. J
Assertion
Ref Expression
riinopn  |-  ( ( J  e.  Top  /\  A  e.  Fin  /\  A. x  e.  A  B  e.  J )  ->  ( X  i^i  |^|_ x  e.  A  B )  e.  J
)
Distinct variable groups:    x, A    x, J    x, X
Allowed substitution hint:    B( x)

Proof of Theorem riinopn
StepHypRef Expression
1 riin0 4167 . . . 4  |-  ( A  =  (/)  ->  ( X  i^i  |^|_ x  e.  A  B )  =  X )
21adantl 454 . . 3  |-  ( ( ( J  e.  Top  /\  A  e.  Fin  /\  A. x  e.  A  B  e.  J )  /\  A  =  (/) )  ->  ( X  i^i  |^|_ x  e.  A  B )  =  X )
3 simpl1 961 . . . 4  |-  ( ( ( J  e.  Top  /\  A  e.  Fin  /\  A. x  e.  A  B  e.  J )  /\  A  =  (/) )  ->  J  e.  Top )
4 1open.1 . . . . 5  |-  X  = 
U. J
54topopn 16984 . . . 4  |-  ( J  e.  Top  ->  X  e.  J )
63, 5syl 16 . . 3  |-  ( ( ( J  e.  Top  /\  A  e.  Fin  /\  A. x  e.  A  B  e.  J )  /\  A  =  (/) )  ->  X  e.  J )
72, 6eqeltrd 2512 . 2  |-  ( ( ( J  e.  Top  /\  A  e.  Fin  /\  A. x  e.  A  B  e.  J )  /\  A  =  (/) )  ->  ( X  i^i  |^|_ x  e.  A  B )  e.  J
)
84eltopss 16985 . . . . . . . 8  |-  ( ( J  e.  Top  /\  B  e.  J )  ->  B  C_  X )
98ex 425 . . . . . . 7  |-  ( J  e.  Top  ->  ( B  e.  J  ->  B 
C_  X ) )
109adantr 453 . . . . . 6  |-  ( ( J  e.  Top  /\  A  e.  Fin )  ->  ( B  e.  J  ->  B  C_  X )
)
1110ralimdv 2787 . . . . 5  |-  ( ( J  e.  Top  /\  A  e.  Fin )  ->  ( A. x  e.  A  B  e.  J  ->  A. x  e.  A  B  C_  X ) )
12113impia 1151 . . . 4  |-  ( ( J  e.  Top  /\  A  e.  Fin  /\  A. x  e.  A  B  e.  J )  ->  A. x  e.  A  B  C_  X
)
13 riinn0 4168 . . . 4  |-  ( ( A. x  e.  A  B  C_  X  /\  A  =/=  (/) )  ->  ( X  i^i  |^|_ x  e.  A  B )  =  |^|_ x  e.  A  B )
1412, 13sylan 459 . . 3  |-  ( ( ( J  e.  Top  /\  A  e.  Fin  /\  A. x  e.  A  B  e.  J )  /\  A  =/=  (/) )  ->  ( X  i^i  |^|_ x  e.  A  B )  =  |^|_ x  e.  A  B )
15 iinopn 16980 . . . . . 6  |-  ( ( J  e.  Top  /\  ( A  e.  Fin  /\  A  =/=  (/)  /\  A. x  e.  A  B  e.  J ) )  ->  |^|_ x  e.  A  B  e.  J )
16153exp2 1172 . . . . 5  |-  ( J  e.  Top  ->  ( A  e.  Fin  ->  ( A  =/=  (/)  ->  ( A. x  e.  A  B  e.  J  ->  |^|_ x  e.  A  B  e.  J ) ) ) )
1716com34 80 . . . 4  |-  ( J  e.  Top  ->  ( A  e.  Fin  ->  ( A. x  e.  A  B  e.  J  ->  ( A  =/=  (/)  ->  |^|_ x  e.  A  B  e.  J ) ) ) )
18173imp1 1167 . . 3  |-  ( ( ( J  e.  Top  /\  A  e.  Fin  /\  A. x  e.  A  B  e.  J )  /\  A  =/=  (/) )  ->  |^|_ x  e.  A  B  e.  J )
1914, 18eqeltrd 2512 . 2  |-  ( ( ( J  e.  Top  /\  A  e.  Fin  /\  A. x  e.  A  B  e.  J )  /\  A  =/=  (/) )  ->  ( X  i^i  |^|_ x  e.  A  B )  e.  J
)
207, 19pm2.61dane 2684 1  |-  ( ( J  e.  Top  /\  A  e.  Fin  /\  A. x  e.  A  B  e.  J )  ->  ( X  i^i  |^|_ x  e.  A  B )  e.  J
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2601   A.wral 2707    i^i cin 3321    C_ wss 3322   (/)c0 3630   U.cuni 4017   |^|_ciin 4096   Fincfn 7112   Topctop 16963
This theorem is referenced by:  rintopn  16987  iuncld  17114
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-iin 4098  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-recs 6636  df-rdg 6671  df-1o 6727  df-oadd 6731  df-er 6908  df-en 7113  df-dom 7114  df-fin 7116  df-top 16968
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