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Theorem fneer 26368
Description: Fineness intersected with its converse is an equivalence relation. (Contributed by Jeff Hankins, 6-Oct-2009.) (Revised by Mario Carneiro, 11-Sep-2015.)
Hypothesis
Ref Expression
fneval.1  |-  .~  =  ( Fne  i^i  `' Fne )
Assertion
Ref Expression
fneer  |-  .~  Er  _V

Proof of Theorem fneer
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 5728 . 2  |-  ( x  =  y  ->  ( topGen `
 x )  =  ( topGen `  y )
)
2 fneval.1 . . . . . 6  |-  .~  =  ( Fne  i^i  `' Fne )
3 inss1 3561 . . . . . 6  |-  ( Fne 
i^i  `' Fne )  C_  Fne
42, 3eqsstri 3378 . . . . 5  |-  .~  C_  Fne
5 fnerel 26347 . . . . 5  |-  Rel  Fne
6 relss 4963 . . . . 5  |-  (  .~  C_ 
Fne  ->  ( Rel  Fne  ->  Rel  .~  ) )
74, 5, 6mp2 9 . . . 4  |-  Rel  .~
8 dfrel4v 5322 . . . 4  |-  ( Rel 
.~ 
<->  .~  =  { <. x ,  y >.  |  x  .~  y } )
97, 8mpbi 200 . . 3  |-  .~  =  { <. x ,  y
>.  |  x  .~  y }
10 vex 2959 . . . . 5  |-  x  e. 
_V
11 vex 2959 . . . . 5  |-  y  e. 
_V
122fneval 26367 . . . . 5  |-  ( ( x  e.  _V  /\  y  e.  _V )  ->  ( x  .~  y  <->  (
topGen `  x )  =  ( topGen `  y )
) )
1310, 11, 12mp2an 654 . . . 4  |-  ( x  .~  y  <->  ( topGen `  x )  =  (
topGen `  y ) )
1413opabbii 4272 . . 3  |-  { <. x ,  y >.  |  x  .~  y }  =  { <. x ,  y
>.  |  ( topGen `  x )  =  (
topGen `  y ) }
159, 14eqtri 2456 . 2  |-  .~  =  { <. x ,  y
>.  |  ( topGen `  x )  =  (
topGen `  y ) }
161, 15eqer 6938 1  |-  .~  Er  _V
Colors of variables: wff set class
Syntax hints:    <-> wb 177    = wceq 1652    e. wcel 1725   _Vcvv 2956    i^i cin 3319    C_ wss 3320   class class class wbr 4212   {copab 4265   `'ccnv 4877   Rel wrel 4883   ` cfv 5454    Er wer 6902   topGenctg 13665   Fnecfne 26339
This theorem is referenced by:  topfneec  26371  topfneec2  26372
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-iota 5418  df-fun 5456  df-fv 5462  df-er 6905  df-topgen 13667  df-fne 26343
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