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Theorem topfneec 26352
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 26349 . . . 4  |-  .~  Er  _V
3 errel 6906 . . . 4  |-  (  .~  Er  _V  ->  Rel  .~  )
42, 3ax-mp 8 . . 3  |-  Rel  .~
5 relelec 6937 . . 3  |-  ( Rel 
.~  ->  ( A  e. 
[ J ]  .~  <->  J  .~  A ) )
64, 5ax-mp 8 . 2  |-  ( A  e.  [ J ]  .~ 
<->  J  .~  A )
74brrelex2i 4911 . . . 4  |-  ( J  .~  A  ->  A  e.  _V )
87a1i 11 . . 3  |-  ( J  e.  Top  ->  ( J  .~  A  ->  A  e.  _V ) )
9 eleq1 2495 . . . . . . 7  |-  ( (
topGen `  A )  =  J  ->  ( ( topGen `
 A )  e. 
Top 
<->  J  e.  Top )
)
109biimparc 474 . . . . . 6  |-  ( ( J  e.  Top  /\  ( topGen `  A )  =  J )  ->  ( topGen `
 A )  e. 
Top )
11 tgclb 17027 . . . . . 6  |-  ( A  e.  TopBases 
<->  ( topGen `  A )  e.  Top )
1210, 11sylibr 204 . . . . 5  |-  ( ( J  e.  Top  /\  ( topGen `  A )  =  J )  ->  A  e. 
TopBases )
13 elex 2956 . . . . 5  |-  ( A  e.  TopBases  ->  A  e.  _V )
1412, 13syl 16 . . . 4  |-  ( ( J  e.  Top  /\  ( topGen `  A )  =  J )  ->  A  e.  _V )
1514ex 424 . . 3  |-  ( J  e.  Top  ->  (
( topGen `  A )  =  J  ->  A  e. 
_V ) )
161fneval 26348 . . . . 5  |-  ( ( J  e.  Top  /\  A  e.  _V )  ->  ( J  .~  A  <->  (
topGen `  J )  =  ( topGen `  A )
) )
17 tgtop 17030 . . . . . . . 8  |-  ( J  e.  Top  ->  ( topGen `
 J )  =  J )
1817eqeq1d 2443 . . . . . . 7  |-  ( J  e.  Top  ->  (
( topGen `  J )  =  ( topGen `  A
)  <->  J  =  ( topGen `
 A ) ) )
19 eqcom 2437 . . . . . . 7  |-  ( J  =  ( topGen `  A
)  <->  ( topGen `  A
)  =  J )
2018, 19syl6bb 253 . . . . . 6  |-  ( J  e.  Top  ->  (
( topGen `  J )  =  ( topGen `  A
)  <->  ( topGen `  A
)  =  J ) )
2120adantr 452 . . . . 5  |-  ( ( J  e.  Top  /\  A  e.  _V )  ->  ( ( topGen `  J
)  =  ( topGen `  A )  <->  ( topGen `  A )  =  J ) )
2216, 21bitrd 245 . . . 4  |-  ( ( J  e.  Top  /\  A  e.  _V )  ->  ( J  .~  A  <->  (
topGen `  A )  =  J ) )
2322ex 424 . . 3  |-  ( J  e.  Top  ->  ( A  e.  _V  ->  ( J  .~  A  <->  ( topGen `  A )  =  J ) ) )
248, 15, 23pm5.21ndd 344 . 2  |-  ( J  e.  Top  ->  ( J  .~  A  <->  ( topGen `  A )  =  J ) )
256, 24syl5bb 249 1  |-  ( J  e.  Top  ->  ( A  e.  [ J ]  .~  <->  ( topGen `  A
)  =  J ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   _Vcvv 2948    i^i cin 3311   class class class wbr 4204   `'ccnv 4869   Rel wrel 4875   ` cfv 5446    Er wer 6894   [cec 6895   topGenctg 13657   Topctop 16950   TopBasesctb 16954   Fnecfne 26320
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fv 5454  df-er 6897  df-ec 6899  df-topgen 13659  df-top 16955  df-bases 16957  df-fne 26324
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