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Theorem riinopn 16944
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 4132 . . . 4  |-  ( A  =  (/)  ->  ( X  i^i  |^|_ x  e.  A  B )  =  X )
21adantl 453 . . 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 960 . . . 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 16942 . . . 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 2486 . 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 16943 . . . . . . . 8  |-  ( ( J  e.  Top  /\  B  e.  J )  ->  B  C_  X )
98ex 424 . . . . . . 7  |-  ( J  e.  Top  ->  ( B  e.  J  ->  B 
C_  X ) )
109adantr 452 . . . . . 6  |-  ( ( J  e.  Top  /\  A  e.  Fin )  ->  ( B  e.  J  ->  B  C_  X )
)
1110ralimdv 2753 . . . . 5  |-  ( ( J  e.  Top  /\  A  e.  Fin )  ->  ( A. x  e.  A  B  e.  J  ->  A. x  e.  A  B  C_  X ) )
12113impia 1150 . . . 4  |-  ( ( J  e.  Top  /\  A  e.  Fin  /\  A. x  e.  A  B  e.  J )  ->  A. x  e.  A  B  C_  X
)
13 riinn0 4133 . . . 4  |-  ( ( A. x  e.  A  B  C_  X  /\  A  =/=  (/) )  ->  ( X  i^i  |^|_ x  e.  A  B )  =  |^|_ x  e.  A  B )
1412, 13sylan 458 . . 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 16938 . . . . . 6  |-  ( ( J  e.  Top  /\  ( A  e.  Fin  /\  A  =/=  (/)  /\  A. x  e.  A  B  e.  J ) )  ->  |^|_ x  e.  A  B  e.  J )
16153exp2 1171 . . . . 5  |-  ( J  e.  Top  ->  ( A  e.  Fin  ->  ( A  =/=  (/)  ->  ( A. x  e.  A  B  e.  J  ->  |^|_ x  e.  A  B  e.  J ) ) ) )
1716com34 79 . . . 4  |-  ( J  e.  Top  ->  ( A  e.  Fin  ->  ( A. x  e.  A  B  e.  J  ->  ( A  =/=  (/)  ->  |^|_ x  e.  A  B  e.  J ) ) ) )
18173imp1 1166 . . 3  |-  ( ( ( J  e.  Top  /\  A  e.  Fin  /\  A. x  e.  A  B  e.  J )  /\  A  =/=  (/) )  ->  |^|_ x  e.  A  B  e.  J )
1914, 18eqeltrd 2486 . 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 2653 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 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2575   A.wral 2674    i^i cin 3287    C_ wss 3288   (/)c0 3596   U.cuni 3983   |^|_ciin 4062   Fincfn 7076   Topctop 16921
This theorem is referenced by:  rintopn  16945  iuncld  17072
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-reu 2681  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-int 4019  df-iun 4063  df-iin 4064  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-recs 6600  df-rdg 6635  df-1o 6691  df-oadd 6695  df-er 6872  df-en 7077  df-dom 7078  df-fin 7080  df-top 16926
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