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Theorem iinopn 16976
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 966 . . 3  |-  ( ( J  e.  Top  /\  ( A  e.  Fin  /\  A  =/=  (/)  /\  A. x  e.  A  B  e.  J ) )  ->  A. x  e.  A  B  e.  J )
2 dfiin2g 4125 . . 3  |-  ( A. x  e.  A  B  e.  J  ->  |^|_ x  e.  A  B  =  |^| { y  |  E. x  e.  A  y  =  B } )
31, 2syl 16 . 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 445 . . 3  |-  ( ( J  e.  Top  /\  ( A  e.  Fin  /\  A  =/=  (/)  /\  A. x  e.  A  B  e.  J ) )  ->  J  e.  Top )
5 eqid 2437 . . . . 5  |-  ( x  e.  A  |->  B )  =  ( x  e.  A  |->  B )
65rnmpt 5117 . . . 4  |-  ran  (
x  e.  A  |->  B )  =  { y  |  E. x  e.  A  y  =  B }
75fmpt 5891 . . . . . 6  |-  ( A. x  e.  A  B  e.  J  <->  ( x  e.  A  |->  B ) : A --> J )
81, 7sylib 190 . . . . 5  |-  ( ( J  e.  Top  /\  ( A  e.  Fin  /\  A  =/=  (/)  /\  A. x  e.  A  B  e.  J ) )  -> 
( x  e.  A  |->  B ) : A --> J )
9 frn 5598 . . . . 5  |-  ( ( x  e.  A  |->  B ) : A --> J  ->  ran  ( x  e.  A  |->  B )  C_  J
)
108, 9syl 16 . . . 4  |-  ( ( J  e.  Top  /\  ( A  e.  Fin  /\  A  =/=  (/)  /\  A. x  e.  A  B  e.  J ) )  ->  ran  ( x  e.  A  |->  B )  C_  J
)
116, 10syl5eqssr 3394 . . 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 5596 . . . . . 6  |-  ( ( x  e.  A  |->  B ) : A --> J  ->  dom  ( x  e.  A  |->  B )  =  A )
138, 12syl 16 . . . . 5  |-  ( ( J  e.  Top  /\  ( A  e.  Fin  /\  A  =/=  (/)  /\  A. x  e.  A  B  e.  J ) )  ->  dom  ( x  e.  A  |->  B )  =  A )
14 simpr2 965 . . . . 5  |-  ( ( J  e.  Top  /\  ( A  e.  Fin  /\  A  =/=  (/)  /\  A. x  e.  A  B  e.  J ) )  ->  A  =/=  (/) )
1513, 14eqnetrd 2620 . . . 4  |-  ( ( J  e.  Top  /\  ( A  e.  Fin  /\  A  =/=  (/)  /\  A. x  e.  A  B  e.  J ) )  ->  dom  ( x  e.  A  |->  B )  =/=  (/) )
16 dm0rn0 5087 . . . . . 6  |-  ( dom  ( x  e.  A  |->  B )  =  (/)  <->  ran  ( x  e.  A  |->  B )  =  (/) )
176eqeq1i 2444 . . . . . 6  |-  ( ran  ( x  e.  A  |->  B )  =  (/)  <->  {
y  |  E. x  e.  A  y  =  B }  =  (/) )
1816, 17bitri 242 . . . . 5  |-  ( dom  ( x  e.  A  |->  B )  =  (/)  <->  {
y  |  E. x  e.  A  y  =  B }  =  (/) )
1918necon3bii 2634 . . . 4  |-  ( dom  ( x  e.  A  |->  B )  =/=  (/)  <->  { y  |  E. x  e.  A  y  =  B }  =/=  (/) )
2015, 19sylib 190 . . 3  |-  ( ( J  e.  Top  /\  ( A  e.  Fin  /\  A  =/=  (/)  /\  A. x  e.  A  B  e.  J ) )  ->  { y  |  E. x  e.  A  y  =  B }  =/=  (/) )
21 simpr1 964 . . . 4  |-  ( ( J  e.  Top  /\  ( A  e.  Fin  /\  A  =/=  (/)  /\  A. x  e.  A  B  e.  J ) )  ->  A  e.  Fin )
22 abrexfi 7408 . . . 4  |-  ( A  e.  Fin  ->  { y  |  E. x  e.  A  y  =  B }  e.  Fin )
2321, 22syl 16 . . 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 16975 . . . 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 420 . . 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 1187 . 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 2511 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 360    /\ w3a 937    = wceq 1653    e. wcel 1726   {cab 2423    =/= wne 2600   A.wral 2706   E.wrex 2707    C_ wss 3321   (/)c0 3629   |^|cint 4051   |^|_ciin 4095    e. cmpt 4267   dom cdm 4879   ran crn 4880   -->wf 5451   Fincfn 7110   Topctop 16959
This theorem is referenced by:  riinopn  16982  subbascn  17319
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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 2418  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702
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 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-reu 2713  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-pss 3337  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-tp 3823  df-op 3824  df-uni 4017  df-int 4052  df-iun 4096  df-iin 4097  df-br 4214  df-opab 4268  df-mpt 4269  df-tr 4304  df-eprel 4495  df-id 4499  df-po 4504  df-so 4505  df-fr 4542  df-we 4544  df-ord 4585  df-on 4586  df-lim 4587  df-suc 4588  df-om 4847  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-1st 6350  df-2nd 6351  df-recs 6634  df-rdg 6669  df-1o 6725  df-oadd 6729  df-er 6906  df-en 7111  df-dom 7112  df-fin 7114  df-top 16964
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