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Theorem topfneec2 30571
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 30567 . 2  |-  ( ( J  e.  Top  /\  K  e.  Top )  ->  ( J  .~  K  <->  (
topGen `  J )  =  ( topGen `  K )
) )
31fneer 30568 . . . 4  |-  .~  Er  _V
43a1i 11 . . 3  |-  ( ( J  e.  Top  /\  K  e.  Top )  ->  .~  Er  _V )
5 elex 3067 . . . 4  |-  ( J  e.  Top  ->  J  e.  _V )
65adantr 463 . . 3  |-  ( ( J  e.  Top  /\  K  e.  Top )  ->  J  e.  _V )
74, 6erth 7392 . 2  |-  ( ( J  e.  Top  /\  K  e.  Top )  ->  ( J  .~  K  <->  [ J ]  .~  =  [ K ]  .~  )
)
8 tgtop 19765 . . 3  |-  ( J  e.  Top  ->  ( topGen `
 J )  =  J )
9 tgtop 19765 . . 3  |-  ( K  e.  Top  ->  ( topGen `
 K )  =  K )
108, 9eqeqan12d 2425 . 2  |-  ( ( J  e.  Top  /\  K  e.  Top )  ->  ( ( topGen `  J
)  =  ( topGen `  K )  <->  J  =  K ) )
112, 7, 103bitr3d 283 1  |-  ( ( J  e.  Top  /\  K  e.  Top )  ->  ( [ J ]  .~  =  [ K ]  .~  <->  J  =  K
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1405    e. wcel 1842   _Vcvv 3058    i^i cin 3412   class class class wbr 4394   `'ccnv 4821   ` cfv 5568    Er wer 7344   [cec 7345   topGenctg 15050   Topctop 19684   Fnecfne 30551
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-iota 5532  df-fun 5570  df-fv 5576  df-er 7347  df-ec 7349  df-topgen 15056  df-top 19689  df-fne 30552
This theorem is referenced by: (None)
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