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Theorem iinopn 19578
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 1002 . . 3  |-  ( ( J  e.  Top  /\  ( A  e.  Fin  /\  A  =/=  (/)  /\  A. x  e.  A  B  e.  J ) )  ->  A. x  e.  A  B  e.  J )
2 dfiin2g 4348 . . 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 455 . . 3  |-  ( ( J  e.  Top  /\  ( A  e.  Fin  /\  A  =/=  (/)  /\  A. x  e.  A  B  e.  J ) )  ->  J  e.  Top )
5 eqid 2454 . . . . 5  |-  ( x  e.  A  |->  B )  =  ( x  e.  A  |->  B )
65rnmpt 5237 . . . 4  |-  ran  (
x  e.  A  |->  B )  =  { y  |  E. x  e.  A  y  =  B }
75fmpt 6028 . . . . . 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 5719 . . . . 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 3534 . . 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 5717 . . . . . 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 1001 . . . . 5  |-  ( ( J  e.  Top  /\  ( A  e.  Fin  /\  A  =/=  (/)  /\  A. x  e.  A  B  e.  J ) )  ->  A  =/=  (/) )
1513, 14eqnetrd 2747 . . . 4  |-  ( ( J  e.  Top  /\  ( A  e.  Fin  /\  A  =/=  (/)  /\  A. x  e.  A  B  e.  J ) )  ->  dom  ( x  e.  A  |->  B )  =/=  (/) )
16 dm0rn0 5208 . . . . . 6  |-  ( dom  ( x  e.  A  |->  B )  =  (/)  <->  ran  ( x  e.  A  |->  B )  =  (/) )
176eqeq1i 2461 . . . . . 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 2722 . . . 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 1000 . . . 4  |-  ( ( J  e.  Top  /\  ( A  e.  Fin  /\  A  =/=  (/)  /\  A. x  e.  A  B  e.  J ) )  ->  A  e.  Fin )
22 abrexfi 7812 . . . 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 19577 . . . 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 427 . . 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 1228 . 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 2542 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 367    /\ w3a 971    = wceq 1398    e. wcel 1823   {cab 2439    =/= wne 2649   A.wral 2804   E.wrex 2805    C_ wss 3461   (/)c0 3783   |^|cint 4271   |^|_ciin 4316    |-> cmpt 4497   dom cdm 4988   ran crn 4989   -->wf 5566   Fincfn 7509   Topctop 19561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-iin 4318  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-en 7510  df-dom 7511  df-fin 7513  df-top 19566
This theorem is referenced by:  riinopn  19584  subbascn  19922
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