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Theorem fneer 29788
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 5866 . 2  |-  ( x  =  y  ->  ( topGen `
 x )  =  ( topGen `  y )
)
2 fneval.1 . . . . . 6  |-  .~  =  ( Fne  i^i  `' Fne )
3 inss1 3718 . . . . . 6  |-  ( Fne 
i^i  `' Fne )  C_  Fne
42, 3eqsstri 3534 . . . . 5  |-  .~  C_  Fne
5 fnerel 29767 . . . . 5  |-  Rel  Fne
6 relss 5090 . . . . 5  |-  (  .~  C_ 
Fne  ->  ( Rel  Fne  ->  Rel  .~  ) )
74, 5, 6mp2 9 . . . 4  |-  Rel  .~
8 dfrel4v 5458 . . . 4  |-  ( Rel 
.~ 
<->  .~  =  { <. x ,  y >.  |  x  .~  y } )
97, 8mpbi 208 . . 3  |-  .~  =  { <. x ,  y
>.  |  x  .~  y }
10 vex 3116 . . . . 5  |-  x  e. 
_V
11 vex 3116 . . . . 5  |-  y  e. 
_V
122fneval 29787 . . . . 5  |-  ( ( x  e.  _V  /\  y  e.  _V )  ->  ( x  .~  y  <->  (
topGen `  x )  =  ( topGen `  y )
) )
1310, 11, 12mp2an 672 . . . 4  |-  ( x  .~  y  <->  ( topGen `  x )  =  (
topGen `  y ) )
1413opabbii 4511 . . 3  |-  { <. x ,  y >.  |  x  .~  y }  =  { <. x ,  y
>.  |  ( topGen `  x )  =  (
topGen `  y ) }
159, 14eqtri 2496 . 2  |-  .~  =  { <. x ,  y
>.  |  ( topGen `  x )  =  (
topGen `  y ) }
161, 15eqer 7344 1  |-  .~  Er  _V
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    = wceq 1379    e. wcel 1767   _Vcvv 3113    i^i cin 3475    C_ wss 3476   class class class wbr 4447   {copab 4504   `'ccnv 4998   Rel wrel 5004   ` cfv 5588    Er wer 7308   topGenctg 14693   Fnecfne 29759
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-iota 5551  df-fun 5590  df-fv 5596  df-er 7311  df-topgen 14699  df-fne 29763
This theorem is referenced by:  topfneec  29791  topfneec2  29792
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