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Theorem topfneec 26291
Description: A cover is equivalent to a topology iff it is a base for that topology. (Contributed by Jeff Hankins, 8-Oct-2009.) (Proof shortened by Mario Carneiro, 11-Sep-2015.)
Hypothesis
Ref Expression
topfneec.1  |-  .~  =  ( Fne  i^i  `' Fne )
Assertion
Ref Expression
topfneec  |-  ( J  e.  Top  ->  ( A  e.  [ J ]  .~  <->  ( topGen `  A
)  =  J ) )

Proof of Theorem topfneec
StepHypRef Expression
1 topfneec.1 . . . . 5  |-  .~  =  ( Fne  i^i  `' Fne )
21fneer 26288 . . . 4  |-  .~  Er  _V
3 errel 6669 . . . 4  |-  (  .~  Er  _V  ->  Rel  .~  )
42, 3ax-mp 8 . . 3  |-  Rel  .~
5 relelec 6700 . . 3  |-  ( Rel 
.~  ->  ( A  e. 
[ J ]  .~  <->  J  .~  A ) )
64, 5ax-mp 8 . 2  |-  ( A  e.  [ J ]  .~ 
<->  J  .~  A )
74brrelex2i 4730 . . . 4  |-  ( J  .~  A  ->  A  e.  _V )
87a1i 10 . . 3  |-  ( J  e.  Top  ->  ( J  .~  A  ->  A  e.  _V ) )
9 eleq1 2343 . . . . . . 7  |-  ( (
topGen `  A )  =  J  ->  ( ( topGen `
 A )  e. 
Top 
<->  J  e.  Top )
)
109biimparc 473 . . . . . 6  |-  ( ( J  e.  Top  /\  ( topGen `  A )  =  J )  ->  ( topGen `
 A )  e. 
Top )
11 tgclb 16708 . . . . . 6  |-  ( A  e.  TopBases 
<->  ( topGen `  A )  e.  Top )
1210, 11sylibr 203 . . . . 5  |-  ( ( J  e.  Top  /\  ( topGen `  A )  =  J )  ->  A  e. 
TopBases )
13 elex 2796 . . . . 5  |-  ( A  e.  TopBases  ->  A  e.  _V )
1412, 13syl 15 . . . 4  |-  ( ( J  e.  Top  /\  ( topGen `  A )  =  J )  ->  A  e.  _V )
1514ex 423 . . 3  |-  ( J  e.  Top  ->  (
( topGen `  A )  =  J  ->  A  e. 
_V ) )
161fneval 26287 . . . . 5  |-  ( ( J  e.  Top  /\  A  e.  _V )  ->  ( J  .~  A  <->  (
topGen `  J )  =  ( topGen `  A )
) )
17 tgtop 16711 . . . . . . . 8  |-  ( J  e.  Top  ->  ( topGen `
 J )  =  J )
1817eqeq1d 2291 . . . . . . 7  |-  ( J  e.  Top  ->  (
( topGen `  J )  =  ( topGen `  A
)  <->  J  =  ( topGen `
 A ) ) )
19 eqcom 2285 . . . . . . 7  |-  ( J  =  ( topGen `  A
)  <->  ( topGen `  A
)  =  J )
2018, 19syl6bb 252 . . . . . 6  |-  ( J  e.  Top  ->  (
( topGen `  J )  =  ( topGen `  A
)  <->  ( topGen `  A
)  =  J ) )
2120adantr 451 . . . . 5  |-  ( ( J  e.  Top  /\  A  e.  _V )  ->  ( ( topGen `  J
)  =  ( topGen `  A )  <->  ( topGen `  A )  =  J ) )
2216, 21bitrd 244 . . . 4  |-  ( ( J  e.  Top  /\  A  e.  _V )  ->  ( J  .~  A  <->  (
topGen `  A )  =  J ) )
2322ex 423 . . 3  |-  ( J  e.  Top  ->  ( A  e.  _V  ->  ( J  .~  A  <->  ( topGen `  A )  =  J ) ) )
248, 15, 23pm5.21ndd 343 . 2  |-  ( J  e.  Top  ->  ( J  .~  A  <->  ( topGen `  A )  =  J ) )
256, 24syl5bb 248 1  |-  ( J  e.  Top  ->  ( A  e.  [ J ]  .~  <->  ( topGen `  A
)  =  J ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788    i^i cin 3151   class class class wbr 4023   `'ccnv 4688   Rel wrel 4694   ` cfv 5255    Er wer 6657   [cec 6658   topGenctg 13342   Topctop 16631   TopBasesctb 16635   Fnecfne 26259
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-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-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-ral 2548  df-rex 2549  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-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  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-fv 5263  df-er 6660  df-ec 6662  df-topgen 13344  df-top 16636  df-bases 16638  df-fne 26263
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