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Theorem alexsub 17739
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 17745 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 451 . . . . . . . 8  |-  ( (
ph  /\  ( f  e.  ( UFil `  X
)  /\  ( J  fLim  f )  =  (/) ) )  ->  X  e. UFL )
3 alexsub.2 . . . . . . . . 9  |-  ( ph  ->  X  =  U. B
)
43adantr 451 . . . . . . . 8  |-  ( (
ph  /\  ( f  e.  ( UFil `  X
)  /\  ( J  fLim  f )  =  (/) ) )  ->  X  =  U. B )
5 alexsub.3 . . . . . . . . 9  |-  ( ph  ->  J  =  ( topGen `  ( fi `  B
) ) )
65adantr 451 . . . . . . . 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 695 . . . . . . . 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 732 . . . . . . . 8  |-  ( (
ph  /\  ( f  e.  ( UFil `  X
)  /\  ( J  fLim  f )  =  (/) ) )  ->  f  e.  ( UFil `  X
) )
10 simprr 733 . . . . . . . 8  |-  ( (
ph  /\  ( f  e.  ( UFil `  X
)  /\  ( J  fLim  f )  =  (/) ) )  ->  ( J  fLim  f )  =  (/) )
112, 4, 6, 8, 9, 10alexsublem 17738 . . . . . . 7  |-  -.  ( ph  /\  ( f  e.  ( UFil `  X
)  /\  ( J  fLim  f )  =  (/) ) )
1211pm2.21i 123 . . . . . 6  |-  ( (
ph  /\  ( f  e.  ( UFil `  X
)  /\  ( J  fLim  f )  =  (/) ) )  ->  -.  ( J  fLim  f )  =  (/) )
1312expr 598 . . . . 5  |-  ( (
ph  /\  f  e.  ( UFil `  X )
)  ->  ( ( J  fLim  f )  =  (/)  ->  -.  ( J  fLim  f )  =  (/) ) )
1413pm2.01d 161 . . . 4  |-  ( (
ph  /\  f  e.  ( UFil `  X )
)  ->  -.  ( J  fLim  f )  =  (/) )
15 df-ne 2448 . . . 4  |-  ( ( J  fLim  f )  =/=  (/)  <->  -.  ( J  fLim  f )  =  (/) )
1614, 15sylibr 203 . . 3  |-  ( (
ph  /\  f  e.  ( UFil `  X )
)  ->  ( J  fLim  f )  =/=  (/) )
1716ralrimiva 2626 . 2  |-  ( ph  ->  A. f  e.  (
UFil `  X )
( J  fLim  f
)  =/=  (/) )
18 fibas 16715 . . . . . 6  |-  ( fi
`  B )  e.  TopBases
19 tgtopon 16709 . . . . . 6  |-  ( ( fi `  B )  e.  TopBases  ->  ( topGen `  ( fi `  B ) )  e.  (TopOn `  U. ( fi `  B ) ) )
2018, 19ax-mp 8 . . . . 5  |-  ( topGen `  ( fi `  B
) )  e.  (TopOn `  U. ( fi `  B ) )
215, 20syl6eqel 2371 . . . 4  |-  ( ph  ->  J  e.  (TopOn `  U. ( fi `  B
) ) )
22 elex 2796 . . . . . . . . . 10  |-  ( X  e. UFL  ->  X  e.  _V )
231, 22syl 15 . . . . . . . . 9  |-  ( ph  ->  X  e.  _V )
243, 23eqeltrrd 2358 . . . . . . . 8  |-  ( ph  ->  U. B  e.  _V )
25 uniexb 4563 . . . . . . . 8  |-  ( B  e.  _V  <->  U. B  e. 
_V )
2624, 25sylibr 203 . . . . . . 7  |-  ( ph  ->  B  e.  _V )
27 fiuni 7181 . . . . . . 7  |-  ( B  e.  _V  ->  U. B  =  U. ( fi `  B ) )
2826, 27syl 15 . . . . . 6  |-  ( ph  ->  U. B  =  U. ( fi `  B ) )
293, 28eqtrd 2315 . . . . 5  |-  ( ph  ->  X  =  U. ( fi `  B ) )
3029fveq2d 5529 . . . 4  |-  ( ph  ->  (TopOn `  X )  =  (TopOn `  U. ( fi
`  B ) ) )
3121, 30eleqtrrd 2360 . . 3  |-  ( ph  ->  J  e.  (TopOn `  X ) )
32 ufilcmp 17727 . . 3  |-  ( ( X  e. UFL  /\  J  e.  (TopOn `  X )
)  ->  ( J  e.  Comp  <->  A. f  e.  (
UFil `  X )
( J  fLim  f
)  =/=  (/) ) )
331, 31, 32syl2anc 642 . 2  |-  ( ph  ->  ( J  e.  Comp  <->  A. f  e.  ( UFil `  X ) ( J 
fLim  f )  =/=  (/) ) )
3417, 33mpbird 223 1  |-  ( ph  ->  J  e.  Comp )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   E.wrex 2544   _Vcvv 2788    i^i cin 3151    C_ wss 3152   (/)c0 3455   ~Pcpw 3625   U.cuni 3827   ` cfv 5255  (class class class)co 5858   Fincfn 6863   ficfi 7164   topGenctg 13342  TopOnctopon 16632   TopBasesctb 16635   Compccmp 17113   UFilcufil 17594  UFLcufl 17595    fLim cflim 17629
This theorem is referenced by:  alexsubb  17740  ptcmplem5  17750
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-fi 7165  df-topgen 13344  df-top 16636  df-bases 16638  df-topon 16639  df-cld 16756  df-ntr 16757  df-cls 16758  df-nei 16835  df-cmp 17114  df-fbas 17520  df-fg 17521  df-fil 17541  df-ufil 17596  df-ufl 17597  df-flim 17634  df-fcls 17636
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