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Theorem topfneec 30596
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 30594 . . . 4  |-  .~  Er  _V
3 errel 7359 . . . 4  |-  (  .~  Er  _V  ->  Rel  .~  )
42, 3ax-mp 5 . . 3  |-  Rel  .~
5 relelec 7391 . . 3  |-  ( Rel 
.~  ->  ( A  e. 
[ J ]  .~  <->  J  .~  A ) )
64, 5ax-mp 5 . 2  |-  ( A  e.  [ J ]  .~ 
<->  J  .~  A )
74brrelex2i 4867 . . . 4  |-  ( J  .~  A  ->  A  e.  _V )
87a1i 11 . . 3  |-  ( J  e.  Top  ->  ( J  .~  A  ->  A  e.  _V ) )
9 eleq1 2476 . . . . . . 7  |-  ( (
topGen `  A )  =  J  ->  ( ( topGen `
 A )  e. 
Top 
<->  J  e.  Top )
)
109biimparc 487 . . . . . 6  |-  ( ( J  e.  Top  /\  ( topGen `  A )  =  J )  ->  ( topGen `
 A )  e. 
Top )
11 tgclb 19766 . . . . . 6  |-  ( A  e.  TopBases 
<->  ( topGen `  A )  e.  Top )
1210, 11sylibr 214 . . . . 5  |-  ( ( J  e.  Top  /\  ( topGen `  A )  =  J )  ->  A  e. 
TopBases )
13 elex 3070 . . . . 5  |-  ( A  e.  TopBases  ->  A  e.  _V )
1412, 13syl 17 . . . 4  |-  ( ( J  e.  Top  /\  ( topGen `  A )  =  J )  ->  A  e.  _V )
1514ex 434 . . 3  |-  ( J  e.  Top  ->  (
( topGen `  A )  =  J  ->  A  e. 
_V ) )
161fneval 30593 . . . . 5  |-  ( ( J  e.  Top  /\  A  e.  _V )  ->  ( J  .~  A  <->  (
topGen `  J )  =  ( topGen `  A )
) )
17 tgtop 19769 . . . . . . . 8  |-  ( J  e.  Top  ->  ( topGen `
 J )  =  J )
1817eqeq1d 2406 . . . . . . 7  |-  ( J  e.  Top  ->  (
( topGen `  J )  =  ( topGen `  A
)  <->  J  =  ( topGen `
 A ) ) )
19 eqcom 2413 . . . . . . 7  |-  ( J  =  ( topGen `  A
)  <->  ( topGen `  A
)  =  J )
2018, 19syl6bb 263 . . . . . 6  |-  ( J  e.  Top  ->  (
( topGen `  J )  =  ( topGen `  A
)  <->  ( topGen `  A
)  =  J ) )
2120adantr 465 . . . . 5  |-  ( ( J  e.  Top  /\  A  e.  _V )  ->  ( ( topGen `  J
)  =  ( topGen `  A )  <->  ( topGen `  A )  =  J ) )
2216, 21bitrd 255 . . . 4  |-  ( ( J  e.  Top  /\  A  e.  _V )  ->  ( J  .~  A  <->  (
topGen `  A )  =  J ) )
2322ex 434 . . 3  |-  ( J  e.  Top  ->  ( A  e.  _V  ->  ( J  .~  A  <->  ( topGen `  A )  =  J ) ) )
248, 15, 23pm5.21ndd 354 . 2  |-  ( J  e.  Top  ->  ( J  .~  A  <->  ( topGen `  A )  =  J ) )
256, 24syl5bb 259 1  |-  ( J  e.  Top  ->  ( A  e.  [ J ]  .~  <->  ( topGen `  A
)  =  J ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 186    /\ wa 369    = wceq 1407    e. wcel 1844   _Vcvv 3061    i^i cin 3415   class class class wbr 4397   `'ccnv 4824   Rel wrel 4830   ` cfv 5571    Er wer 7347   [cec 7348   topGenctg 15054   Topctop 19688   TopBasesctb 19692   Fnecfne 30577
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3063  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-op 3981  df-uni 4194  df-iun 4275  df-br 4398  df-opab 4456  df-mpt 4457  df-id 4740  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-iota 5535  df-fun 5573  df-fv 5579  df-er 7350  df-ec 7352  df-topgen 15060  df-top 19693  df-bases 19695  df-fne 30578
This theorem is referenced by: (None)
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