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Theorem topfneec 26394
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 26391 . . . 4  |-  .~  Er  _V
3 errel 6685 . . . 4  |-  (  .~  Er  _V  ->  Rel  .~  )
42, 3ax-mp 8 . . 3  |-  Rel  .~
5 relelec 6716 . . 3  |-  ( Rel 
.~  ->  ( A  e. 
[ J ]  .~  <->  J  .~  A ) )
64, 5ax-mp 8 . 2  |-  ( A  e.  [ J ]  .~ 
<->  J  .~  A )
74brrelex2i 4746 . . . 4  |-  ( J  .~  A  ->  A  e.  _V )
87a1i 10 . . 3  |-  ( J  e.  Top  ->  ( J  .~  A  ->  A  e.  _V ) )
9 eleq1 2356 . . . . . . 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 16724 . . . . . 6  |-  ( A  e.  TopBases 
<->  ( topGen `  A )  e.  Top )
1210, 11sylibr 203 . . . . 5  |-  ( ( J  e.  Top  /\  ( topGen `  A )  =  J )  ->  A  e. 
TopBases )
13 elex 2809 . . . . 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 26390 . . . . 5  |-  ( ( J  e.  Top  /\  A  e.  _V )  ->  ( J  .~  A  <->  (
topGen `  J )  =  ( topGen `  A )
) )
17 tgtop 16727 . . . . . . . 8  |-  ( J  e.  Top  ->  ( topGen `
 J )  =  J )
1817eqeq1d 2304 . . . . . . 7  |-  ( J  e.  Top  ->  (
( topGen `  J )  =  ( topGen `  A
)  <->  J  =  ( topGen `
 A ) ) )
19 eqcom 2298 . . . . . . 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 1632    e. wcel 1696   _Vcvv 2801    i^i cin 3164   class class class wbr 4039   `'ccnv 4704   Rel wrel 4710   ` cfv 5271    Er wer 6673   [cec 6674   topGenctg 13358   Topctop 16647   TopBasesctb 16651   Fnecfne 26362
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-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-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-ral 2561  df-rex 2562  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-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  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-fv 5279  df-er 6676  df-ec 6678  df-topgen 13360  df-top 16652  df-bases 16654  df-fne 26366
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