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Theorem chintcli 25925
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 25819 . . . . 5  |-  CH  C_  SH
42, 3sstri 3513 . . . 4  |-  A  C_  SH
51simpri 462 . . . 4  |-  A  =/=  (/)
64, 5pm3.2i 455 . . 3  |-  ( A 
C_  SH  /\  A  =/=  (/) )
76shintcli 25923 . 2  |-  |^| A  e.  SH
82sseli 3500 . . . . . . . 8  |-  ( y  e.  A  ->  y  e.  CH )
9 vex 3116 . . . . . . . . . . 11  |-  x  e. 
_V
109chlimi 25828 . . . . . . . . . 10  |-  ( ( y  e.  CH  /\  f : NN --> y  /\  f  ~~>v  x )  ->  x  e.  y )
11103exp 1195 . . . . . . . . 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 2872 . . . . 5  |-  ( f 
~~>v  x  ->  ( A. y  e.  A  f : NN --> y  ->  A. y  e.  A  x  e.  y ) )
165fint 5762 . . . . 5  |-  ( f : NN --> |^| A  <->  A. y  e.  A  f : NN --> y )
179elint2 4289 . . . . 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 1602 . 2  |-  A. f A. x ( ( f : NN --> |^| A  /\  f  ~~>v  x )  ->  x  e.  |^| A )
21 isch2 25817 . 2  |-  ( |^| A  e.  CH  <->  ( |^| A  e.  SH  /\  A. f A. x ( ( f : NN --> |^| A  /\  f  ~~>v  x )  ->  x  e.  |^| A ) ) )
227, 20, 21mpbir2an 918 1  |-  |^| A  e.  CH
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369   A.wal 1377    e. wcel 1767    =/= wne 2662   A.wral 2814    C_ wss 3476   (/)c0 3785   |^|cint 4282   class class class wbr 4447   -->wf 5582   NNcn 10532    ~~>v chli 25520   SHcsh 25521   CHcch 25522
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-i2m1 9556  ax-1ne0 9557  ax-rrecex 9560  ax-cnre 9561  ax-hilex 25592  ax-hfvadd 25593  ax-hv0cl 25596  ax-hfvmul 25598
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-recs 7039  df-rdg 7073  df-map 7419  df-nn 10533  df-sh 25800  df-ch 25815
This theorem is referenced by:  chintcl  25926  chincli  26054
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