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Theorem alexsub 18068
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 18074 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 18067 . . . . . . 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 2668 . . 3  |-  ( (
ph  /\  f  e.  ( UFil `  X )
)  ->  ( J  fLim  f )  =/=  (/) )
1615ralrimiva 2781 . 2  |-  ( ph  ->  A. f  e.  (
UFil `  X )
( J  fLim  f
)  =/=  (/) )
17 fibas 17034 . . . . . 6  |-  ( fi
`  B )  e.  TopBases
18 tgtopon 17028 . . . . . 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 2523 . . . 4  |-  ( ph  ->  J  e.  (TopOn `  U. ( fi `  B
) ) )
21 elex 2956 . . . . . . . . . 10  |-  ( X  e. UFL  ->  X  e.  _V )
221, 21syl 16 . . . . . . . . 9  |-  ( ph  ->  X  e.  _V )
233, 22eqeltrrd 2510 . . . . . . . 8  |-  ( ph  ->  U. B  e.  _V )
24 uniexb 4744 . . . . . . . 8  |-  ( B  e.  _V  <->  U. B  e. 
_V )
2523, 24sylibr 204 . . . . . . 7  |-  ( ph  ->  B  e.  _V )
26 fiuni 7425 . . . . . . 7  |-  ( B  e.  _V  ->  U. B  =  U. ( fi `  B ) )
2725, 26syl 16 . . . . . 6  |-  ( ph  ->  U. B  =  U. ( fi `  B ) )
283, 27eqtrd 2467 . . . . 5  |-  ( ph  ->  X  =  U. ( fi `  B ) )
2928fveq2d 5724 . . . 4  |-  ( ph  ->  (TopOn `  X )  =  (TopOn `  U. ( fi
`  B ) ) )
3020, 29eleqtrrd 2512 . . 3  |-  ( ph  ->  J  e.  (TopOn `  X ) )
31 ufilcmp 18056 . . 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 1652    e. wcel 1725    =/= wne 2598   A.wral 2697   E.wrex 2698   _Vcvv 2948    i^i cin 3311    C_ wss 3312   (/)c0 3620   ~Pcpw 3791   U.cuni 4007   ` cfv 5446  (class class class)co 6073   Fincfn 7101   ficfi 7407   topGenctg 13657  TopOnctopon 16951   TopBasesctb 16954   Compccmp 17441   UFilcufil 17923  UFLcufl 17924    fLim cflim 17958
This theorem is referenced by:  alexsubb  18069  ptcmplem5  18079
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
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-nel 2601  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-iin 4088  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-1st 6341  df-2nd 6342  df-recs 6625  df-rdg 6660  df-1o 6716  df-2o 6717  df-oadd 6720  df-er 6897  df-map 7012  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-fi 7408  df-topgen 13659  df-fbas 16691  df-fg 16692  df-top 16955  df-bases 16957  df-topon 16958  df-cld 17075  df-ntr 17076  df-cls 17077  df-nei 17154  df-cmp 17442  df-fil 17870  df-ufil 17925  df-ufl 17926  df-flim 17963  df-fcls 17965
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