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Theorem chintcli 24879
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 458 . . . . 5  |-  A  C_  CH
3 chsssh 24773 . . . . 5  |-  CH  C_  SH
42, 3sstri 3466 . . . 4  |-  A  C_  SH
51simpri 462 . . . 4  |-  A  =/=  (/)
64, 5pm3.2i 455 . . 3  |-  ( A 
C_  SH  /\  A  =/=  (/) )
76shintcli 24877 . 2  |-  |^| A  e.  SH
82sseli 3453 . . . . . . . 8  |-  ( y  e.  A  ->  y  e.  CH )
9 vex 3074 . . . . . . . . . . 11  |-  x  e. 
_V
109chlimi 24782 . . . . . . . . . 10  |-  ( ( y  e.  CH  /\  f : NN --> y  /\  f  ~~>v  x )  ->  x  e.  y )
11103exp 1187 . . . . . . . . 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 32 . . . . . . 7  |-  ( f 
~~>v  x  ->  ( y  e.  A  ->  ( f : NN --> y  ->  x  e.  y )
) )
1413imp 429 . . . . . 6  |-  ( ( f  ~~>v  x  /\  y  e.  A )  ->  (
f : NN --> y  ->  x  e.  y )
)
1514ralimdva 2827 . . . . 5  |-  ( f 
~~>v  x  ->  ( A. y  e.  A  f : NN --> y  ->  A. y  e.  A  x  e.  y ) )
165fint 5691 . . . . 5  |-  ( f : NN --> |^| A  <->  A. y  e.  A  f : NN --> y )
179elint2 4236 . . . . 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 430 . . 3  |-  ( ( f : NN --> |^| A  /\  f  ~~>v  x )  ->  x  e.  |^| A )
2019gen2 1593 . 2  |-  A. f A. x ( ( f : NN --> |^| A  /\  f  ~~>v  x )  ->  x  e.  |^| A )
21 isch2 24771 . 2  |-  ( |^| A  e.  CH  <->  ( |^| A  e.  SH  /\  A. f A. x ( ( f : NN --> |^| A  /\  f  ~~>v  x )  ->  x  e.  |^| A ) ) )
227, 20, 21mpbir2an 911 1  |-  |^| A  e.  CH
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369   A.wal 1368    e. wcel 1758    =/= wne 2644   A.wral 2795    C_ wss 3429   (/)c0 3738   |^|cint 4229   class class class wbr 4393   -->wf 5515   NNcn 10426    ~~>v chli 24474   SHcsh 24475   CHcch 24476
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-cnex 9442  ax-resscn 9443  ax-1cn 9444  ax-icn 9445  ax-addcl 9446  ax-addrcl 9447  ax-mulcl 9448  ax-mulrcl 9449  ax-i2m1 9454  ax-1ne0 9455  ax-rrecex 9458  ax-cnre 9459  ax-hilex 24546  ax-hfvadd 24547  ax-hv0cl 24550  ax-hfvmul 24552
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-pss 3445  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-tp 3983  df-op 3985  df-uni 4193  df-int 4230  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-tr 4487  df-eprel 4733  df-id 4737  df-po 4742  df-so 4743  df-fr 4780  df-we 4782  df-ord 4823  df-on 4824  df-lim 4825  df-suc 4826  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-om 6580  df-recs 6935  df-rdg 6969  df-map 7319  df-nn 10427  df-sh 24754  df-ch 24769
This theorem is referenced by:  chintcl  24880  chincli  25008
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