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Theorem alexsub 17999
Description: The Alexander Subbase Theorem: If  B is a subbase for the topology  J, and any cover taken from  B has a finite subcover, then the generated topology is compact. This proof uses the ultrafilter lemma; see alexsubALT 18005 for a proof using Zorn's lemma. (Contributed by Jeff Hankins, 24-Jan-2010.) (Revised by Mario Carneiro, 26-Aug-2015.)
Hypotheses
Ref Expression
alexsub.1  |-  ( ph  ->  X  e. UFL )
alexsub.2  |-  ( ph  ->  X  =  U. B
)
alexsub.3  |-  ( ph  ->  J  =  ( topGen `  ( fi `  B
) ) )
alexsub.4  |-  ( (
ph  /\  ( x  C_  B  /\  X  = 
U. x ) )  ->  E. y  e.  ( ~P x  i^i  Fin ) X  =  U. y )
Assertion
Ref Expression
alexsub  |-  ( ph  ->  J  e.  Comp )
Distinct variable groups:    x, y, B    x, J, y    ph, x, y    x, X, y

Proof of Theorem alexsub
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 alexsub.1 . . . . . . . . 9  |-  ( ph  ->  X  e. UFL )
21adantr 452 . . . . . . . 8  |-  ( (
ph  /\  ( f  e.  ( UFil `  X
)  /\  ( J  fLim  f )  =  (/) ) )  ->  X  e. UFL )
3 alexsub.2 . . . . . . . . 9  |-  ( ph  ->  X  =  U. B
)
43adantr 452 . . . . . . . 8  |-  ( (
ph  /\  ( f  e.  ( UFil `  X
)  /\  ( J  fLim  f )  =  (/) ) )  ->  X  =  U. B )
5 alexsub.3 . . . . . . . . 9  |-  ( ph  ->  J  =  ( topGen `  ( fi `  B
) ) )
65adantr 452 . . . . . . . 8  |-  ( (
ph  /\  ( f  e.  ( UFil `  X
)  /\  ( J  fLim  f )  =  (/) ) )  ->  J  =  ( topGen `  ( fi `  B ) ) )
7 alexsub.4 . . . . . . . . 9  |-  ( (
ph  /\  ( x  C_  B  /\  X  = 
U. x ) )  ->  E. y  e.  ( ~P x  i^i  Fin ) X  =  U. y )
87adantlr 696 . . . . . . . 8  |-  ( ( ( ph  /\  (
f  e.  ( UFil `  X )  /\  ( J  fLim  f )  =  (/) ) )  /\  (
x  C_  B  /\  X  =  U. x
) )  ->  E. y  e.  ( ~P x  i^i 
Fin ) X  = 
U. y )
9 simprl 733 . . . . . . . 8  |-  ( (
ph  /\  ( f  e.  ( UFil `  X
)  /\  ( J  fLim  f )  =  (/) ) )  ->  f  e.  ( UFil `  X
) )
10 simprr 734 . . . . . . . 8  |-  ( (
ph  /\  ( f  e.  ( UFil `  X
)  /\  ( J  fLim  f )  =  (/) ) )  ->  ( J  fLim  f )  =  (/) )
112, 4, 6, 8, 9, 10alexsublem 17998 . . . . . . 7  |-  -.  ( ph  /\  ( f  e.  ( UFil `  X
)  /\  ( J  fLim  f )  =  (/) ) )
1211pm2.21i 125 . . . . . 6  |-  ( (
ph  /\  ( f  e.  ( UFil `  X
)  /\  ( J  fLim  f )  =  (/) ) )  ->  -.  ( J  fLim  f )  =  (/) )
1312expr 599 . . . . 5  |-  ( (
ph  /\  f  e.  ( UFil `  X )
)  ->  ( ( J  fLim  f )  =  (/)  ->  -.  ( J  fLim  f )  =  (/) ) )
1413pm2.01d 163 . . . 4  |-  ( (
ph  /\  f  e.  ( UFil `  X )
)  ->  -.  ( J  fLim  f )  =  (/) )
1514neneqad 2622 . . 3  |-  ( (
ph  /\  f  e.  ( UFil `  X )
)  ->  ( J  fLim  f )  =/=  (/) )
1615ralrimiva 2734 . 2  |-  ( ph  ->  A. f  e.  (
UFil `  X )
( J  fLim  f
)  =/=  (/) )
17 fibas 16967 . . . . . 6  |-  ( fi
`  B )  e.  TopBases
18 tgtopon 16961 . . . . . 6  |-  ( ( fi `  B )  e.  TopBases  ->  ( topGen `  ( fi `  B ) )  e.  (TopOn `  U. ( fi `  B ) ) )
1917, 18ax-mp 8 . . . . 5  |-  ( topGen `  ( fi `  B
) )  e.  (TopOn `  U. ( fi `  B ) )
205, 19syl6eqel 2477 . . . 4  |-  ( ph  ->  J  e.  (TopOn `  U. ( fi `  B
) ) )
21 elex 2909 . . . . . . . . . 10  |-  ( X  e. UFL  ->  X  e.  _V )
221, 21syl 16 . . . . . . . . 9  |-  ( ph  ->  X  e.  _V )
233, 22eqeltrrd 2464 . . . . . . . 8  |-  ( ph  ->  U. B  e.  _V )
24 uniexb 4694 . . . . . . . 8  |-  ( B  e.  _V  <->  U. B  e. 
_V )
2523, 24sylibr 204 . . . . . . 7  |-  ( ph  ->  B  e.  _V )
26 fiuni 7370 . . . . . . 7  |-  ( B  e.  _V  ->  U. B  =  U. ( fi `  B ) )
2725, 26syl 16 . . . . . 6  |-  ( ph  ->  U. B  =  U. ( fi `  B ) )
283, 27eqtrd 2421 . . . . 5  |-  ( ph  ->  X  =  U. ( fi `  B ) )
2928fveq2d 5674 . . . 4  |-  ( ph  ->  (TopOn `  X )  =  (TopOn `  U. ( fi
`  B ) ) )
3020, 29eleqtrrd 2466 . . 3  |-  ( ph  ->  J  e.  (TopOn `  X ) )
31 ufilcmp 17987 . . 3  |-  ( ( X  e. UFL  /\  J  e.  (TopOn `  X )
)  ->  ( J  e.  Comp  <->  A. f  e.  (
UFil `  X )
( J  fLim  f
)  =/=  (/) ) )
321, 30, 31syl2anc 643 . 2  |-  ( ph  ->  ( J  e.  Comp  <->  A. f  e.  ( UFil `  X ) ( J 
fLim  f )  =/=  (/) ) )
3316, 32mpbird 224 1  |-  ( ph  ->  J  e.  Comp )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717    =/= wne 2552   A.wral 2651   E.wrex 2652   _Vcvv 2901    i^i cin 3264    C_ wss 3265   (/)c0 3573   ~Pcpw 3744   U.cuni 3959   ` cfv 5396  (class class class)co 6022   Fincfn 7047   ficfi 7352   topGenctg 13594  TopOnctopon 16884   TopBasesctb 16887   Compccmp 17373   UFilcufil 17854  UFLcufl 17855    fLim cflim 17889
This theorem is referenced by:  alexsubb  18000  ptcmplem5  18010
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
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-nel 2555  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-iin 4040  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-1st 6290  df-2nd 6291  df-recs 6571  df-rdg 6606  df-1o 6662  df-2o 6663  df-oadd 6666  df-er 6843  df-map 6958  df-en 7048  df-dom 7049  df-sdom 7050  df-fin 7051  df-fi 7353  df-topgen 13596  df-fbas 16625  df-fg 16626  df-top 16888  df-bases 16890  df-topon 16891  df-cld 17008  df-ntr 17009  df-cls 17010  df-nei 17087  df-cmp 17374  df-fil 17801  df-ufil 17856  df-ufl 17857  df-flim 17894  df-fcls 17896
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