MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  iinopn Structured version   Unicode version

Theorem iinopn 18520
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 996 . . 3  |-  ( ( J  e.  Top  /\  ( A  e.  Fin  /\  A  =/=  (/)  /\  A. x  e.  A  B  e.  J ) )  ->  A. x  e.  A  B  e.  J )
2 dfiin2g 4208 . . 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 2443 . . . . 5  |-  ( x  e.  A  |->  B )  =  ( x  e.  A  |->  B )
65rnmpt 5090 . . . 4  |-  ran  (
x  e.  A  |->  B )  =  { y  |  E. x  e.  A  y  =  B }
75fmpt 5869 . . . . . 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 5570 . . . . 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 3406 . . 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 5568 . . . . . 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 995 . . . . 5  |-  ( ( J  e.  Top  /\  ( A  e.  Fin  /\  A  =/=  (/)  /\  A. x  e.  A  B  e.  J ) )  ->  A  =/=  (/) )
1513, 14eqnetrd 2631 . . . 4  |-  ( ( J  e.  Top  /\  ( A  e.  Fin  /\  A  =/=  (/)  /\  A. x  e.  A  B  e.  J ) )  ->  dom  ( x  e.  A  |->  B )  =/=  (/) )
16 dm0rn0 5061 . . . . . 6  |-  ( dom  ( x  e.  A  |->  B )  =  (/)  <->  ran  ( x  e.  A  |->  B )  =  (/) )
176eqeq1i 2450 . . . . . 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 2645 . . . 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 994 . . . 4  |-  ( ( J  e.  Top  /\  ( A  e.  Fin  /\  A  =/=  (/)  /\  A. x  e.  A  B  e.  J ) )  ->  A  e.  Fin )
22 abrexfi 7616 . . . 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 18519 . . . 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 1220 . 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 2517 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 965    = wceq 1369    e. wcel 1756   {cab 2429    =/= wne 2611   A.wral 2720   E.wrex 2721    C_ wss 3333   (/)c0 3642   |^|cint 4133   |^|_ciin 4177    e. cmpt 4355   dom cdm 4845   ran crn 4846   -->wf 5419   Fincfn 7315   Topctop 18503
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-reu 2727  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-iin 4179  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  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 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-recs 6837  df-rdg 6871  df-1o 6925  df-oadd 6929  df-er 7106  df-en 7316  df-dom 7317  df-fin 7319  df-top 18508
This theorem is referenced by:  riinopn  18526  subbascn  18863
  Copyright terms: Public domain W3C validator