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Theorem alexsub 18029
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 18035 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 18028 . . . . . . 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 2637 . . 3  |-  ( (
ph  /\  f  e.  ( UFil `  X )
)  ->  ( J  fLim  f )  =/=  (/) )
1615ralrimiva 2749 . 2  |-  ( ph  ->  A. f  e.  (
UFil `  X )
( J  fLim  f
)  =/=  (/) )
17 fibas 16997 . . . . . 6  |-  ( fi
`  B )  e.  TopBases
18 tgtopon 16991 . . . . . 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 2492 . . . 4  |-  ( ph  ->  J  e.  (TopOn `  U. ( fi `  B
) ) )
21 elex 2924 . . . . . . . . . 10  |-  ( X  e. UFL  ->  X  e.  _V )
221, 21syl 16 . . . . . . . . 9  |-  ( ph  ->  X  e.  _V )
233, 22eqeltrrd 2479 . . . . . . . 8  |-  ( ph  ->  U. B  e.  _V )
24 uniexb 4711 . . . . . . . 8  |-  ( B  e.  _V  <->  U. B  e. 
_V )
2523, 24sylibr 204 . . . . . . 7  |-  ( ph  ->  B  e.  _V )
26 fiuni 7391 . . . . . . 7  |-  ( B  e.  _V  ->  U. B  =  U. ( fi `  B ) )
2725, 26syl 16 . . . . . 6  |-  ( ph  ->  U. B  =  U. ( fi `  B ) )
283, 27eqtrd 2436 . . . . 5  |-  ( ph  ->  X  =  U. ( fi `  B ) )
2928fveq2d 5691 . . . 4  |-  ( ph  ->  (TopOn `  X )  =  (TopOn `  U. ( fi
`  B ) ) )
3020, 29eleqtrrd 2481 . . 3  |-  ( ph  ->  J  e.  (TopOn `  X ) )
31 ufilcmp 18017 . . 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 1721    =/= wne 2567   A.wral 2666   E.wrex 2667   _Vcvv 2916    i^i cin 3279    C_ wss 3280   (/)c0 3588   ~Pcpw 3759   U.cuni 3975   ` cfv 5413  (class class class)co 6040   Fincfn 7068   ficfi 7373   topGenctg 13620  TopOnctopon 16914   TopBasesctb 16917   Compccmp 17403   UFilcufil 17884  UFLcufl 17885    fLim cflim 17919
This theorem is referenced by:  alexsubb  18030  ptcmplem5  18040
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-map 6979  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-fi 7374  df-topgen 13622  df-fbas 16654  df-fg 16655  df-top 16918  df-bases 16920  df-topon 16921  df-cld 17038  df-ntr 17039  df-cls 17040  df-nei 17117  df-cmp 17404  df-fil 17831  df-ufil 17886  df-ufl 17887  df-flim 17924  df-fcls 17926
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