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Theorem fneer 26288
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 5525 . 2  |-  ( x  =  y  ->  ( topGen `
 x )  =  ( topGen `  y )
)
2 fneval.1 . . . . . 6  |-  .~  =  ( Fne  i^i  `' Fne )
3 inss1 3389 . . . . . 6  |-  ( Fne 
i^i  `' Fne )  C_  Fne
42, 3eqsstri 3208 . . . . 5  |-  .~  C_  Fne
5 fnerel 26267 . . . . 5  |-  Rel  Fne
6 relss 4775 . . . . 5  |-  (  .~  C_ 
Fne  ->  ( Rel  Fne  ->  Rel  .~  ) )
74, 5, 6mp2 17 . . . 4  |-  Rel  .~
8 dfrel4v 5125 . . . 4  |-  ( Rel 
.~ 
<->  .~  =  { <. x ,  y >.  |  x  .~  y } )
97, 8mpbi 199 . . 3  |-  .~  =  { <. x ,  y
>.  |  x  .~  y }
10 vex 2791 . . . . 5  |-  x  e. 
_V
11 vex 2791 . . . . 5  |-  y  e. 
_V
122fneval 26287 . . . . 5  |-  ( ( x  e.  _V  /\  y  e.  _V )  ->  ( x  .~  y  <->  (
topGen `  x )  =  ( topGen `  y )
) )
1310, 11, 12mp2an 653 . . . 4  |-  ( x  .~  y  <->  ( topGen `  x )  =  (
topGen `  y ) )
1413opabbii 4083 . . 3  |-  { <. x ,  y >.  |  x  .~  y }  =  { <. x ,  y
>.  |  ( topGen `  x )  =  (
topGen `  y ) }
159, 14eqtri 2303 . 2  |-  .~  =  { <. x ,  y
>.  |  ( topGen `  x )  =  (
topGen `  y ) }
161, 15eqer 6693 1  |-  .~  Er  _V
Colors of variables: wff set class
Syntax hints:    <-> wb 176    = wceq 1623    e. wcel 1684   _Vcvv 2788    i^i cin 3151    C_ wss 3152   class class class wbr 4023   {copab 4076   `'ccnv 4688   Rel wrel 4694   ` cfv 5255    Er wer 6657   topGenctg 13342   Fnecfne 26259
This theorem is referenced by:  topfneec  26291  topfneec2  26292
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-iota 5219  df-fun 5257  df-fv 5263  df-er 6660  df-topgen 13344  df-fne 26263
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