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Theorem iinopn 19257
Description: The intersection of a nonempty 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 1004 . . 3  |-  ( ( J  e.  Top  /\  ( A  e.  Fin  /\  A  =/=  (/)  /\  A. x  e.  A  B  e.  J ) )  ->  A. x  e.  A  B  e.  J )
2 dfiin2g 4363 . . 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 457 . . 3  |-  ( ( J  e.  Top  /\  ( A  e.  Fin  /\  A  =/=  (/)  /\  A. x  e.  A  B  e.  J ) )  ->  J  e.  Top )
5 eqid 2467 . . . . 5  |-  ( x  e.  A  |->  B )  =  ( x  e.  A  |->  B )
65rnmpt 5253 . . . 4  |-  ran  (
x  e.  A  |->  B )  =  { y  |  E. x  e.  A  y  =  B }
75fmpt 6052 . . . . . 6  |-  ( A. x  e.  A  B  e.  J  <->  ( x  e.  A  |->  B ) : A --> J )
81, 7sylib 196 . . . . 5  |-  ( ( J  e.  Top  /\  ( A  e.  Fin  /\  A  =/=  (/)  /\  A. x  e.  A  B  e.  J ) )  -> 
( x  e.  A  |->  B ) : A --> J )
9 frn 5742 . . . . 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 3554 . . 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 5740 . . . . . 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 1003 . . . . 5  |-  ( ( J  e.  Top  /\  ( A  e.  Fin  /\  A  =/=  (/)  /\  A. x  e.  A  B  e.  J ) )  ->  A  =/=  (/) )
1513, 14eqnetrd 2760 . . . 4  |-  ( ( J  e.  Top  /\  ( A  e.  Fin  /\  A  =/=  (/)  /\  A. x  e.  A  B  e.  J ) )  ->  dom  ( x  e.  A  |->  B )  =/=  (/) )
16 dm0rn0 5224 . . . . . 6  |-  ( dom  ( x  e.  A  |->  B )  =  (/)  <->  ran  ( x  e.  A  |->  B )  =  (/) )
176eqeq1i 2474 . . . . . 6  |-  ( ran  ( x  e.  A  |->  B )  =  (/)  <->  {
y  |  E. x  e.  A  y  =  B }  =  (/) )
1816, 17bitri 249 . . . . 5  |-  ( dom  ( x  e.  A  |->  B )  =  (/)  <->  {
y  |  E. x  e.  A  y  =  B }  =  (/) )
1918necon3bii 2735 . . . 4  |-  ( dom  ( x  e.  A  |->  B )  =/=  (/)  <->  { y  |  E. x  e.  A  y  =  B }  =/=  (/) )
2015, 19sylib 196 . . 3  |-  ( ( J  e.  Top  /\  ( A  e.  Fin  /\  A  =/=  (/)  /\  A. x  e.  A  B  e.  J ) )  ->  { y  |  E. x  e.  A  y  =  B }  =/=  (/) )
21 simpr1 1002 . . . 4  |-  ( ( J  e.  Top  /\  ( A  e.  Fin  /\  A  =/=  (/)  /\  A. x  e.  A  B  e.  J ) )  ->  A  e.  Fin )
22 abrexfi 7830 . . . 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 19256 . . . 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 429 . . 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 1230 . 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 2555 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   {cab 2452    =/= wne 2662   A.wral 2817   E.wrex 2818    C_ wss 3481   (/)c0 3790   |^|cint 4287   |^|_ciin 4331    |-> cmpt 4510   dom cdm 5004   ran crn 5005   -->wf 5589   Fincfn 7526   Topctop 19240
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4251  df-int 4288  df-iun 4332  df-iin 4333  df-br 4453  df-opab 4511  df-mpt 4512  df-tr 4546  df-eprel 4796  df-id 4800  df-po 4805  df-so 4806  df-fr 4843  df-we 4845  df-ord 4886  df-on 4887  df-lim 4888  df-suc 4889  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-ov 6297  df-oprab 6298  df-mpt2 6299  df-om 6695  df-1st 6794  df-2nd 6795  df-recs 7052  df-rdg 7086  df-1o 7140  df-oadd 7144  df-er 7321  df-en 7527  df-dom 7528  df-fin 7530  df-top 19245
This theorem is referenced by:  riinopn  19263  subbascn  19600
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