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Theorem fneer 30594
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 5851 . 2  |-  ( x  =  y  ->  ( topGen `
 x )  =  ( topGen `  y )
)
2 fneval.1 . . . . . 6  |-  .~  =  ( Fne  i^i  `' Fne )
3 inss1 3661 . . . . . 6  |-  ( Fne 
i^i  `' Fne )  C_  Fne
42, 3eqsstri 3474 . . . . 5  |-  .~  C_  Fne
5 fnerel 30579 . . . . 5  |-  Rel  Fne
6 relss 4913 . . . . 5  |-  (  .~  C_ 
Fne  ->  ( Rel  Fne  ->  Rel  .~  ) )
74, 5, 6mp2 9 . . . 4  |-  Rel  .~
8 dfrel4v 5277 . . . 4  |-  ( Rel 
.~ 
<->  .~  =  { <. x ,  y >.  |  x  .~  y } )
97, 8mpbi 210 . . 3  |-  .~  =  { <. x ,  y
>.  |  x  .~  y }
10 vex 3064 . . . . 5  |-  x  e. 
_V
11 vex 3064 . . . . 5  |-  y  e. 
_V
122fneval 30593 . . . . 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 4461 . . 3  |-  { <. x ,  y >.  |  x  .~  y }  =  { <. x ,  y
>.  |  ( topGen `  x )  =  (
topGen `  y ) }
159, 14eqtri 2433 . 2  |-  .~  =  { <. x ,  y
>.  |  ( topGen `  x )  =  (
topGen `  y ) }
161, 15eqer 7383 1  |-  .~  Er  _V
Colors of variables: wff setvar class
Syntax hints:    <-> wb 186    = wceq 1407    e. wcel 1844   _Vcvv 3061    i^i cin 3415    C_ wss 3416   class class class wbr 4397   {copab 4454   `'ccnv 4824   Rel wrel 4830   ` cfv 5571    Er wer 7347   topGenctg 15054   Fnecfne 30577
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3063  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-op 3981  df-uni 4194  df-iun 4275  df-br 4398  df-opab 4456  df-mpt 4457  df-id 4740  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-iota 5535  df-fun 5573  df-fv 5579  df-er 7350  df-topgen 15060  df-fne 30578
This theorem is referenced by:  topfneec  30596  topfneec2  30597
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