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Theorem topfneec2 26372
Description: A topology is precisely identified with its equivalence class. (Contributed by Jeff Hankins, 12-Oct-2009.)
Hypothesis
Ref Expression
topfneec2.1  |-  .~  =  ( Fne  i^i  `' Fne )
Assertion
Ref Expression
topfneec2  |-  ( ( J  e.  Top  /\  K  e.  Top )  ->  ( [ J ]  .~  =  [ K ]  .~  <->  J  =  K
) )

Proof of Theorem topfneec2
StepHypRef Expression
1 topfneec2.1 . . 3  |-  .~  =  ( Fne  i^i  `' Fne )
21fneval 26367 . 2  |-  ( ( J  e.  Top  /\  K  e.  Top )  ->  ( J  .~  K  <->  (
topGen `  J )  =  ( topGen `  K )
) )
31fneer 26368 . . . 4  |-  .~  Er  _V
43a1i 11 . . 3  |-  ( ( J  e.  Top  /\  K  e.  Top )  ->  .~  Er  _V )
5 elex 2964 . . . 4  |-  ( J  e.  Top  ->  J  e.  _V )
65adantr 452 . . 3  |-  ( ( J  e.  Top  /\  K  e.  Top )  ->  J  e.  _V )
74, 6erth 6949 . 2  |-  ( ( J  e.  Top  /\  K  e.  Top )  ->  ( J  .~  K  <->  [ J ]  .~  =  [ K ]  .~  )
)
8 tgtop 17038 . . 3  |-  ( J  e.  Top  ->  ( topGen `
 J )  =  J )
9 tgtop 17038 . . 3  |-  ( K  e.  Top  ->  ( topGen `
 K )  =  K )
108, 9eqeqan12d 2451 . 2  |-  ( ( J  e.  Top  /\  K  e.  Top )  ->  ( ( topGen `  J
)  =  ( topGen `  K )  <->  J  =  K ) )
112, 7, 103bitr3d 275 1  |-  ( ( J  e.  Top  /\  K  e.  Top )  ->  ( [ J ]  .~  =  [ K ]  .~  <->  J  =  K
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   _Vcvv 2956    i^i cin 3319   class class class wbr 4212   `'ccnv 4877   ` cfv 5454    Er wer 6902   [cec 6903   topGenctg 13665   Topctop 16958   Fnecfne 26339
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 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fv 5462  df-er 6905  df-ec 6907  df-topgen 13667  df-top 16963  df-fne 26343
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