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Theorem alexsub 17755
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 17761 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 17754 . . . . . . 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 2461 . . . 4  |-  ( ( J  fLim  f )  =/=  (/)  <->  -.  ( J  fLim  f )  =  (/) )
1614, 15sylibr 203 . . 3  |-  ( (
ph  /\  f  e.  ( UFil `  X )
)  ->  ( J  fLim  f )  =/=  (/) )
1716ralrimiva 2639 . 2  |-  ( ph  ->  A. f  e.  (
UFil `  X )
( J  fLim  f
)  =/=  (/) )
18 fibas 16731 . . . . . 6  |-  ( fi
`  B )  e.  TopBases
19 tgtopon 16725 . . . . . 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 2384 . . . 4  |-  ( ph  ->  J  e.  (TopOn `  U. ( fi `  B
) ) )
22 elex 2809 . . . . . . . . . 10  |-  ( X  e. UFL  ->  X  e.  _V )
231, 22syl 15 . . . . . . . . 9  |-  ( ph  ->  X  e.  _V )
243, 23eqeltrrd 2371 . . . . . . . 8  |-  ( ph  ->  U. B  e.  _V )
25 uniexb 4579 . . . . . . . 8  |-  ( B  e.  _V  <->  U. B  e. 
_V )
2624, 25sylibr 203 . . . . . . 7  |-  ( ph  ->  B  e.  _V )
27 fiuni 7197 . . . . . . 7  |-  ( B  e.  _V  ->  U. B  =  U. ( fi `  B ) )
2826, 27syl 15 . . . . . 6  |-  ( ph  ->  U. B  =  U. ( fi `  B ) )
293, 28eqtrd 2328 . . . . 5  |-  ( ph  ->  X  =  U. ( fi `  B ) )
3029fveq2d 5545 . . . 4  |-  ( ph  ->  (TopOn `  X )  =  (TopOn `  U. ( fi
`  B ) ) )
3121, 30eleqtrrd 2373 . . 3  |-  ( ph  ->  J  e.  (TopOn `  X ) )
32 ufilcmp 17743 . . 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 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   E.wrex 2557   _Vcvv 2801    i^i cin 3164    C_ wss 3165   (/)c0 3468   ~Pcpw 3638   U.cuni 3843   ` cfv 5271  (class class class)co 5874   Fincfn 6879   ficfi 7180   topGenctg 13358  TopOnctopon 16648   TopBasesctb 16651   Compccmp 17129   UFilcufil 17610  UFLcufl 17611    fLim cflim 17645
This theorem is referenced by:  alexsubb  17756  ptcmplem5  17766
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-fi 7181  df-topgen 13360  df-top 16652  df-bases 16654  df-topon 16655  df-cld 16772  df-ntr 16773  df-cls 16774  df-nei 16851  df-cmp 17130  df-fbas 17536  df-fg 17537  df-fil 17557  df-ufil 17612  df-ufl 17613  df-flim 17650  df-fcls 17652
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