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Theorem topfneec2 26292
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 26287 . 2  |-  ( ( J  e.  Top  /\  K  e.  Top )  ->  ( J  .~  K  <->  (
topGen `  J )  =  ( topGen `  K )
) )
31fneer 26288 . . . 4  |-  .~  Er  _V
43a1i 10 . . 3  |-  ( ( J  e.  Top  /\  K  e.  Top )  ->  .~  Er  _V )
5 elex 2796 . . . 4  |-  ( J  e.  Top  ->  J  e.  _V )
65adantr 451 . . 3  |-  ( ( J  e.  Top  /\  K  e.  Top )  ->  J  e.  _V )
74, 6erth 6704 . 2  |-  ( ( J  e.  Top  /\  K  e.  Top )  ->  ( J  .~  K  <->  [ J ]  .~  =  [ K ]  .~  )
)
8 tgtop 16711 . . 3  |-  ( J  e.  Top  ->  ( topGen `
 J )  =  J )
9 tgtop 16711 . . 3  |-  ( K  e.  Top  ->  ( topGen `
 K )  =  K )
108, 9eqeqan12d 2298 . 2  |-  ( ( J  e.  Top  /\  K  e.  Top )  ->  ( ( topGen `  J
)  =  ( topGen `  K )  <->  J  =  K ) )
112, 7, 103bitr3d 274 1  |-  ( ( J  e.  Top  /\  K  e.  Top )  ->  ( [ J ]  .~  =  [ K ]  .~  <->  J  =  K
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788    i^i cin 3151   class class class wbr 4023   `'ccnv 4688   ` cfv 5255    Er wer 6657   [cec 6658   topGenctg 13342   Topctop 16631   Fnecfne 26259
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fv 5263  df-er 6660  df-ec 6662  df-topgen 13344  df-top 16636  df-fne 26263
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