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Theorem iinopn 16664
Description: The intersection of a non-empty finite family of open sets is open. (Contributed by Mario Carneiro, 14-Sep-2014.)
Assertion
Ref Expression
iinopn  |-  ( ( J  e.  Top  /\  ( A  e.  Fin  /\  A  =/=  (/)  /\  A. x  e.  A  B  e.  J ) )  ->  |^|_ x  e.  A  B  e.  J )
Distinct variable groups:    x, A    x, J
Allowed substitution hint:    B( x)

Proof of Theorem iinopn
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 simpr3 963 . . 3  |-  ( ( J  e.  Top  /\  ( A  e.  Fin  /\  A  =/=  (/)  /\  A. x  e.  A  B  e.  J ) )  ->  A. x  e.  A  B  e.  J )
2 dfiin2g 3952 . . 3  |-  ( A. x  e.  A  B  e.  J  ->  |^|_ x  e.  A  B  =  |^| { y  |  E. x  e.  A  y  =  B } )
31, 2syl 15 . 2  |-  ( ( J  e.  Top  /\  ( A  e.  Fin  /\  A  =/=  (/)  /\  A. x  e.  A  B  e.  J ) )  ->  |^|_ x  e.  A  B  =  |^| { y  |  E. x  e.  A  y  =  B }
)
4 simpl 443 . . 3  |-  ( ( J  e.  Top  /\  ( A  e.  Fin  /\  A  =/=  (/)  /\  A. x  e.  A  B  e.  J ) )  ->  J  e.  Top )
5 eqid 2296 . . . . 5  |-  ( x  e.  A  |->  B )  =  ( x  e.  A  |->  B )
65rnmpt 4941 . . . 4  |-  ran  (
x  e.  A  |->  B )  =  { y  |  E. x  e.  A  y  =  B }
75fmpt 5697 . . . . . 6  |-  ( A. x  e.  A  B  e.  J  <->  ( x  e.  A  |->  B ) : A --> J )
81, 7sylib 188 . . . . 5  |-  ( ( J  e.  Top  /\  ( A  e.  Fin  /\  A  =/=  (/)  /\  A. x  e.  A  B  e.  J ) )  -> 
( x  e.  A  |->  B ) : A --> J )
9 frn 5411 . . . . 5  |-  ( ( x  e.  A  |->  B ) : A --> J  ->  ran  ( x  e.  A  |->  B )  C_  J
)
108, 9syl 15 . . . 4  |-  ( ( J  e.  Top  /\  ( A  e.  Fin  /\  A  =/=  (/)  /\  A. x  e.  A  B  e.  J ) )  ->  ran  ( x  e.  A  |->  B )  C_  J
)
116, 10syl5eqssr 3236 . . 3  |-  ( ( J  e.  Top  /\  ( A  e.  Fin  /\  A  =/=  (/)  /\  A. x  e.  A  B  e.  J ) )  ->  { y  |  E. x  e.  A  y  =  B }  C_  J
)
12 fdm 5409 . . . . . 6  |-  ( ( x  e.  A  |->  B ) : A --> J  ->  dom  ( x  e.  A  |->  B )  =  A )
138, 12syl 15 . . . . 5  |-  ( ( J  e.  Top  /\  ( A  e.  Fin  /\  A  =/=  (/)  /\  A. x  e.  A  B  e.  J ) )  ->  dom  ( x  e.  A  |->  B )  =  A )
14 simpr2 962 . . . . 5  |-  ( ( J  e.  Top  /\  ( A  e.  Fin  /\  A  =/=  (/)  /\  A. x  e.  A  B  e.  J ) )  ->  A  =/=  (/) )
1513, 14eqnetrd 2477 . . . 4  |-  ( ( J  e.  Top  /\  ( A  e.  Fin  /\  A  =/=  (/)  /\  A. x  e.  A  B  e.  J ) )  ->  dom  ( x  e.  A  |->  B )  =/=  (/) )
16 dm0rn0 4911 . . . . . 6  |-  ( dom  ( x  e.  A  |->  B )  =  (/)  <->  ran  ( x  e.  A  |->  B )  =  (/) )
176eqeq1i 2303 . . . . . 6  |-  ( ran  ( x  e.  A  |->  B )  =  (/)  <->  {
y  |  E. x  e.  A  y  =  B }  =  (/) )
1816, 17bitri 240 . . . . 5  |-  ( dom  ( x  e.  A  |->  B )  =  (/)  <->  {
y  |  E. x  e.  A  y  =  B }  =  (/) )
1918necon3bii 2491 . . . 4  |-  ( dom  ( x  e.  A  |->  B )  =/=  (/)  <->  { y  |  E. x  e.  A  y  =  B }  =/=  (/) )
2015, 19sylib 188 . . 3  |-  ( ( J  e.  Top  /\  ( A  e.  Fin  /\  A  =/=  (/)  /\  A. x  e.  A  B  e.  J ) )  ->  { y  |  E. x  e.  A  y  =  B }  =/=  (/) )
21 simpr1 961 . . . 4  |-  ( ( J  e.  Top  /\  ( A  e.  Fin  /\  A  =/=  (/)  /\  A. x  e.  A  B  e.  J ) )  ->  A  e.  Fin )
22 abrexfi 7172 . . . 4  |-  ( A  e.  Fin  ->  { y  |  E. x  e.  A  y  =  B }  e.  Fin )
2321, 22syl 15 . . 3  |-  ( ( J  e.  Top  /\  ( A  e.  Fin  /\  A  =/=  (/)  /\  A. x  e.  A  B  e.  J ) )  ->  { y  |  E. x  e.  A  y  =  B }  e.  Fin )
24 fiinopn 16663 . . . 4  |-  ( J  e.  Top  ->  (
( { y  |  E. x  e.  A  y  =  B }  C_  J  /\  { y  |  E. x  e.  A  y  =  B }  =/=  (/)  /\  {
y  |  E. x  e.  A  y  =  B }  e.  Fin )  ->  |^| { y  |  E. x  e.  A  y  =  B }  e.  J ) )
2524imp 418 . . 3  |-  ( ( J  e.  Top  /\  ( { y  |  E. x  e.  A  y  =  B }  C_  J  /\  { y  |  E. x  e.  A  y  =  B }  =/=  (/)  /\  {
y  |  E. x  e.  A  y  =  B }  e.  Fin ) )  ->  |^| { y  |  E. x  e.  A  y  =  B }  e.  J )
264, 11, 20, 23, 25syl13anc 1184 . 2  |-  ( ( J  e.  Top  /\  ( A  e.  Fin  /\  A  =/=  (/)  /\  A. x  e.  A  B  e.  J ) )  ->  |^| { y  |  E. x  e.  A  y  =  B }  e.  J
)
273, 26eqeltrd 2370 1  |-  ( ( J  e.  Top  /\  ( A  e.  Fin  /\  A  =/=  (/)  /\  A. x  e.  A  B  e.  J ) )  ->  |^|_ x  e.  A  B  e.  J )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   {cab 2282    =/= wne 2459   A.wral 2556   E.wrex 2557    C_ wss 3165   (/)c0 3468   |^|cint 3878   |^|_ciin 3922    e. cmpt 4093   dom cdm 4705   ran crn 4706   -->wf 5267   Fincfn 6879   Topctop 16647
This theorem is referenced by:  riinopn  16670  subbascn  17000
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-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-en 6880  df-dom 6881  df-fin 6883  df-top 16652
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