Users' Mathboxes Mathbox for Jeff Hankins < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  fneer Structured version   Unicode version

Theorem fneer 28586
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 5712 . 2  |-  ( x  =  y  ->  ( topGen `
 x )  =  ( topGen `  y )
)
2 fneval.1 . . . . . 6  |-  .~  =  ( Fne  i^i  `' Fne )
3 inss1 3591 . . . . . 6  |-  ( Fne 
i^i  `' Fne )  C_  Fne
42, 3eqsstri 3407 . . . . 5  |-  .~  C_  Fne
5 fnerel 28565 . . . . 5  |-  Rel  Fne
6 relss 4948 . . . . 5  |-  (  .~  C_ 
Fne  ->  ( Rel  Fne  ->  Rel  .~  ) )
74, 5, 6mp2 9 . . . 4  |-  Rel  .~
8 dfrel4v 5310 . . . 4  |-  ( Rel 
.~ 
<->  .~  =  { <. x ,  y >.  |  x  .~  y } )
97, 8mpbi 208 . . 3  |-  .~  =  { <. x ,  y
>.  |  x  .~  y }
10 vex 2996 . . . . 5  |-  x  e. 
_V
11 vex 2996 . . . . 5  |-  y  e. 
_V
122fneval 28585 . . . . 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 4377 . . 3  |-  { <. x ,  y >.  |  x  .~  y }  =  { <. x ,  y
>.  |  ( topGen `  x )  =  (
topGen `  y ) }
159, 14eqtri 2463 . 2  |-  .~  =  { <. x ,  y
>.  |  ( topGen `  x )  =  (
topGen `  y ) }
161, 15eqer 7155 1  |-  .~  Er  _V
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    = wceq 1369    e. wcel 1756   _Vcvv 2993    i^i cin 3348    C_ wss 3349   class class class wbr 4313   {copab 4370   `'ccnv 4860   Rel wrel 4866   ` cfv 5439    Er wer 7119   topGenctg 14397   Fnecfne 28557
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2741  df-rex 2742  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-op 3905  df-uni 4113  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-id 4657  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-iota 5402  df-fun 5441  df-fv 5447  df-er 7122  df-topgen 14403  df-fne 28561
This theorem is referenced by:  topfneec  28589  topfneec2  28590
  Copyright terms: Public domain W3C validator