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Theorem topfneec 29752
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 29749 . . . 4  |-  .~  Er  _V
3 errel 7312 . . . 4  |-  (  .~  Er  _V  ->  Rel  .~  )
42, 3ax-mp 5 . . 3  |-  Rel  .~
5 relelec 7344 . . 3  |-  ( Rel 
.~  ->  ( A  e. 
[ J ]  .~  <->  J  .~  A ) )
64, 5ax-mp 5 . 2  |-  ( A  e.  [ J ]  .~ 
<->  J  .~  A )
74brrelex2i 5035 . . . 4  |-  ( J  .~  A  ->  A  e.  _V )
87a1i 11 . . 3  |-  ( J  e.  Top  ->  ( J  .~  A  ->  A  e.  _V ) )
9 eleq1 2534 . . . . . . 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 19233 . . . . . 6  |-  ( A  e.  TopBases 
<->  ( topGen `  A )  e.  Top )
1210, 11sylibr 212 . . . . 5  |-  ( ( J  e.  Top  /\  ( topGen `  A )  =  J )  ->  A  e. 
TopBases )
13 elex 3117 . . . . 5  |-  ( A  e.  TopBases  ->  A  e.  _V )
1412, 13syl 16 . . . 4  |-  ( ( J  e.  Top  /\  ( topGen `  A )  =  J )  ->  A  e.  _V )
1514ex 434 . . 3  |-  ( J  e.  Top  ->  (
( topGen `  A )  =  J  ->  A  e. 
_V ) )
161fneval 29748 . . . . 5  |-  ( ( J  e.  Top  /\  A  e.  _V )  ->  ( J  .~  A  <->  (
topGen `  J )  =  ( topGen `  A )
) )
17 tgtop 19236 . . . . . . . 8  |-  ( J  e.  Top  ->  ( topGen `
 J )  =  J )
1817eqeq1d 2464 . . . . . . 7  |-  ( J  e.  Top  ->  (
( topGen `  J )  =  ( topGen `  A
)  <->  J  =  ( topGen `
 A ) ) )
19 eqcom 2471 . . . . . . 7  |-  ( J  =  ( topGen `  A
)  <->  ( topGen `  A
)  =  J )
2018, 19syl6bb 261 . . . . . 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 253 . . . 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 257 1  |-  ( J  e.  Top  ->  ( A  e.  [ J ]  .~  <->  ( topGen `  A
)  =  J ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1374    e. wcel 1762   _Vcvv 3108    i^i cin 3470   class class class wbr 4442   `'ccnv 4993   Rel wrel 4999   ` cfv 5581    Er wer 7300   [cec 7301   topGenctg 14684   Topctop 19156   TopBasesctb 19160   Fnecfne 29720
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-id 4790  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fv 5589  df-er 7303  df-ec 7305  df-topgen 14690  df-top 19161  df-bases 19163  df-fne 29724
This theorem is referenced by: (None)
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