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Theorem chintcli 22683
Description: The intersection of a non-empty set of closed subspaces is a closed subspace. (Contributed by NM, 14-Oct-1999.) (New usage is discouraged.)
Hypothesis
Ref Expression
chintcl.1  |-  ( A 
C_  CH  /\  A  =/=  (/) )
Assertion
Ref Expression
chintcli  |-  |^| A  e.  CH

Proof of Theorem chintcli
Dummy variables  x  f  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 chintcl.1 . . . . . 6  |-  ( A 
C_  CH  /\  A  =/=  (/) )
21simpli 445 . . . . 5  |-  A  C_  CH
3 chsssh 22578 . . . . 5  |-  CH  C_  SH
42, 3sstri 3302 . . . 4  |-  A  C_  SH
51simpri 449 . . . 4  |-  A  =/=  (/)
64, 5pm3.2i 442 . . 3  |-  ( A 
C_  SH  /\  A  =/=  (/) )
76shintcli 22681 . 2  |-  |^| A  e.  SH
82sseli 3289 . . . . . . . 8  |-  ( y  e.  A  ->  y  e.  CH )
9 vex 2904 . . . . . . . . . . 11  |-  x  e. 
_V
109chlimi 22587 . . . . . . . . . 10  |-  ( ( y  e.  CH  /\  f : NN --> y  /\  f  ~~>v  x )  ->  x  e.  y )
11103exp 1152 . . . . . . . . 9  |-  ( y  e.  CH  ->  (
f : NN --> y  -> 
( f  ~~>v  x  ->  x  e.  y )
) )
1211com3r 75 . . . . . . . 8  |-  ( f 
~~>v  x  ->  ( y  e.  CH  ->  ( f : NN --> y  ->  x  e.  y ) ) )
138, 12syl5 30 . . . . . . 7  |-  ( f 
~~>v  x  ->  ( y  e.  A  ->  ( f : NN --> y  ->  x  e.  y )
) )
1413imp 419 . . . . . 6  |-  ( ( f  ~~>v  x  /\  y  e.  A )  ->  (
f : NN --> y  ->  x  e.  y )
)
1514ralimdva 2729 . . . . 5  |-  ( f 
~~>v  x  ->  ( A. y  e.  A  f : NN --> y  ->  A. y  e.  A  x  e.  y ) )
165fint 5564 . . . . 5  |-  ( f : NN --> |^| A  <->  A. y  e.  A  f : NN --> y )
179elint2 4001 . . . . 5  |-  ( x  e.  |^| A  <->  A. y  e.  A  x  e.  y )
1815, 16, 173imtr4g 262 . . . 4  |-  ( f 
~~>v  x  ->  ( f : NN --> |^| A  ->  x  e.  |^| A ) )
1918impcom 420 . . 3  |-  ( ( f : NN --> |^| A  /\  f  ~~>v  x )  ->  x  e.  |^| A )
2019gen2 1553 . 2  |-  A. f A. x ( ( f : NN --> |^| A  /\  f  ~~>v  x )  ->  x  e.  |^| A )
21 isch2 22576 . 2  |-  ( |^| A  e.  CH  <->  ( |^| A  e.  SH  /\  A. f A. x ( ( f : NN --> |^| A  /\  f  ~~>v  x )  ->  x  e.  |^| A ) ) )
227, 20, 21mpbir2an 887 1  |-  |^| A  e.  CH
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359   A.wal 1546    e. wcel 1717    =/= wne 2552   A.wral 2651    C_ wss 3265   (/)c0 3573   |^|cint 3994   class class class wbr 4155   -->wf 5392   NNcn 9934    ~~>v chli 22280   SHcsh 22281   CHcch 22282
This theorem is referenced by:  chintcl  22684  chincli  22812
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643  ax-cnex 8981  ax-resscn 8982  ax-1cn 8983  ax-icn 8984  ax-addcl 8985  ax-addrcl 8986  ax-mulcl 8987  ax-mulrcl 8988  ax-i2m1 8993  ax-1ne0 8994  ax-rrecex 8997  ax-cnre 8998  ax-hilex 22352  ax-hfvadd 22353  ax-hv0cl 22356  ax-hfvmul 22358
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-reu 2658  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-pss 3281  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-tp 3767  df-op 3768  df-uni 3960  df-int 3995  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-tr 4246  df-eprel 4437  df-id 4441  df-po 4446  df-so 4447  df-fr 4484  df-we 4486  df-ord 4527  df-on 4528  df-lim 4529  df-suc 4530  df-om 4788  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-recs 6571  df-rdg 6606  df-map 6958  df-nn 9935  df-sh 22559  df-ch 22574
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