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Theorem chintcli 26649
Description: The intersection of a nonempty 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 456 . . . . 5  |-  A  C_  CH
3 chsssh 26543 . . . . 5  |-  CH  C_  SH
42, 3sstri 3450 . . . 4  |-  A  C_  SH
51simpri 460 . . . 4  |-  A  =/=  (/)
64, 5pm3.2i 453 . . 3  |-  ( A 
C_  SH  /\  A  =/=  (/) )
76shintcli 26647 . 2  |-  |^| A  e.  SH
82sseli 3437 . . . . . . . 8  |-  ( y  e.  A  ->  y  e.  CH )
9 vex 3061 . . . . . . . . . . 11  |-  x  e. 
_V
109chlimi 26552 . . . . . . . . . 10  |-  ( ( y  e.  CH  /\  f : NN --> y  /\  f  ~~>v  x )  ->  x  e.  y )
11103exp 1196 . . . . . . . . 9  |-  ( y  e.  CH  ->  (
f : NN --> y  -> 
( f  ~~>v  x  ->  x  e.  y )
) )
1211com3r 79 . . . . . . . 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 427 . . . . . 6  |-  ( ( f  ~~>v  x  /\  y  e.  A )  ->  (
f : NN --> y  ->  x  e.  y )
)
1514ralimdva 2811 . . . . 5  |-  ( f 
~~>v  x  ->  ( A. y  e.  A  f : NN --> y  ->  A. y  e.  A  x  e.  y ) )
165fint 5746 . . . . 5  |-  ( f : NN --> |^| A  <->  A. y  e.  A  f : NN --> y )
179elint2 4233 . . . . 5  |-  ( x  e.  |^| A  <->  A. y  e.  A  x  e.  y )
1815, 16, 173imtr4g 270 . . . 4  |-  ( f 
~~>v  x  ->  ( f : NN --> |^| A  ->  x  e.  |^| A ) )
1918impcom 428 . . 3  |-  ( ( f : NN --> |^| A  /\  f  ~~>v  x )  ->  x  e.  |^| A )
2019gen2 1640 . 2  |-  A. f A. x ( ( f : NN --> |^| A  /\  f  ~~>v  x )  ->  x  e.  |^| A )
21 isch2 26541 . 2  |-  ( |^| A  e.  CH  <->  ( |^| A  e.  SH  /\  A. f A. x ( ( f : NN --> |^| A  /\  f  ~~>v  x )  ->  x  e.  |^| A ) ) )
227, 20, 21mpbir2an 921 1  |-  |^| A  e.  CH
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367   A.wal 1403    e. wcel 1842    =/= wne 2598   A.wral 2753    C_ wss 3413   (/)c0 3737   |^|cint 4226   class class class wbr 4394   -->wf 5564   NNcn 10575    ~~>v chli 26244   SHcsh 26245   CHcch 26246
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573  ax-cnex 9577  ax-resscn 9578  ax-1cn 9579  ax-icn 9580  ax-addcl 9581  ax-addrcl 9582  ax-mulcl 9583  ax-mulrcl 9584  ax-i2m1 9589  ax-1ne0 9590  ax-rrecex 9593  ax-cnre 9594  ax-hilex 26316  ax-hfvadd 26317  ax-hv0cl 26320  ax-hfvmul 26322
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-reu 2760  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-int 4227  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-we 4783  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-pred 5366  df-ord 5412  df-on 5413  df-lim 5414  df-suc 5415  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6683  df-wrecs 7012  df-recs 7074  df-rdg 7112  df-map 7458  df-nn 10576  df-sh 26524  df-ch 26539
This theorem is referenced by:  chintcl  26650  chincli  26778
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