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Theorem chintcli 22825
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 22720 . . . . 5  |-  CH  C_  SH
42, 3sstri 3349 . . . 4  |-  A  C_  SH
51simpri 449 . . . 4  |-  A  =/=  (/)
64, 5pm3.2i 442 . . 3  |-  ( A 
C_  SH  /\  A  =/=  (/) )
76shintcli 22823 . 2  |-  |^| A  e.  SH
82sseli 3336 . . . . . . . 8  |-  ( y  e.  A  ->  y  e.  CH )
9 vex 2951 . . . . . . . . . . 11  |-  x  e. 
_V
109chlimi 22729 . . . . . . . . . 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 2776 . . . . 5  |-  ( f 
~~>v  x  ->  ( A. y  e.  A  f : NN --> y  ->  A. y  e.  A  x  e.  y ) )
165fint 5614 . . . . 5  |-  ( f : NN --> |^| A  <->  A. y  e.  A  f : NN --> y )
179elint2 4049 . . . . 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 1556 . 2  |-  A. f A. x ( ( f : NN --> |^| A  /\  f  ~~>v  x )  ->  x  e.  |^| A )
21 isch2 22718 . 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 1549    e. wcel 1725    =/= wne 2598   A.wral 2697    C_ wss 3312   (/)c0 3620   |^|cint 4042   class class class wbr 4204   -->wf 5442   NNcn 9992    ~~>v chli 22422   SHcsh 22423   CHcch 22424
This theorem is referenced by:  chintcl  22826  chincli  22954
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-i2m1 9050  ax-1ne0 9051  ax-rrecex 9054  ax-cnre 9055  ax-hilex 22494  ax-hfvadd 22495  ax-hv0cl 22498  ax-hfvmul 22500
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-recs 6625  df-rdg 6660  df-map 7012  df-nn 9993  df-sh 22701  df-ch 22716
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