Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dffr5 Unicode version

Theorem dffr5 24936
Description: A quantifier free definition of a well-founded relationship. (Contributed by Scott Fenton, 11-Apr-2011.)
Assertion
Ref Expression
dffr5  |-  ( R  Fr  A  <->  ( ~P A  \  { (/) } ) 
C_  ran  (  _E  \  (  _E  o.  `' R ) ) )

Proof of Theorem dffr5
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eldif 3248 . . . . 5  |-  ( x  e.  ( ~P A  \  { (/) } )  <->  ( x  e.  ~P A  /\  -.  x  e.  { (/) } ) )
2 vex 2876 . . . . . . 7  |-  x  e. 
_V
32elpw 3720 . . . . . 6  |-  ( x  e.  ~P A  <->  x  C_  A
)
4 elsn 3744 . . . . . . 7  |-  ( x  e.  { (/) }  <->  x  =  (/) )
54necon3bbii 2560 . . . . . 6  |-  ( -.  x  e.  { (/) }  <-> 
x  =/=  (/) )
63, 5anbi12i 678 . . . . 5  |-  ( ( x  e.  ~P A  /\  -.  x  e.  { (/)
} )  <->  ( x  C_  A  /\  x  =/=  (/) ) )
71, 6bitri 240 . . . 4  |-  ( x  e.  ( ~P A  \  { (/) } )  <->  ( x  C_  A  /\  x  =/=  (/) ) )
8 brdif 4173 . . . . . . 7  |-  ( y (  _E  \  (  _E  o.  `' R ) ) x  <->  ( y  _E  x  /\  -.  y
(  _E  o.  `' R ) x ) )
9 epel 4411 . . . . . . . 8  |-  ( y  _E  x  <->  y  e.  x )
10 vex 2876 . . . . . . . . . . 11  |-  y  e. 
_V
1110, 2coep 24934 . . . . . . . . . 10  |-  ( y (  _E  o.  `' R ) x  <->  E. z  e.  x  y `' R z )
12 vex 2876 . . . . . . . . . . . 12  |-  z  e. 
_V
1310, 12brcnv 4967 . . . . . . . . . . 11  |-  ( y `' R z  <->  z R
y )
1413rexbii 2653 . . . . . . . . . 10  |-  ( E. z  e.  x  y `' R z  <->  E. z  e.  x  z R
y )
15 dfrex2 2641 . . . . . . . . . 10  |-  ( E. z  e.  x  z R y  <->  -.  A. z  e.  x  -.  z R y )
1611, 14, 153bitrri 263 . . . . . . . . 9  |-  ( -. 
A. z  e.  x  -.  z R y  <->  y (  _E  o.  `' R ) x )
1716con1bii 321 . . . . . . . 8  |-  ( -.  y (  _E  o.  `' R ) x  <->  A. z  e.  x  -.  z R y )
189, 17anbi12i 678 . . . . . . 7  |-  ( ( y  _E  x  /\  -.  y (  _E  o.  `' R ) x )  <-> 
( y  e.  x  /\  A. z  e.  x  -.  z R y ) )
198, 18bitri 240 . . . . . 6  |-  ( y (  _E  \  (  _E  o.  `' R ) ) x  <->  ( y  e.  x  /\  A. z  e.  x  -.  z R y ) )
2019exbii 1587 . . . . 5  |-  ( E. y  y (  _E 
\  (  _E  o.  `' R ) ) x  <->  E. y ( y  e.  x  /\  A. z  e.  x  -.  z R y ) )
212elrn 5022 . . . . 5  |-  ( x  e.  ran  (  _E 
\  (  _E  o.  `' R ) )  <->  E. y 
y (  _E  \ 
(  _E  o.  `' R ) ) x )
22 df-rex 2634 . . . . 5  |-  ( E. y  e.  x  A. z  e.  x  -.  z R y  <->  E. y
( y  e.  x  /\  A. z  e.  x  -.  z R y ) )
2320, 21, 223bitr4i 268 . . . 4  |-  ( x  e.  ran  (  _E 
\  (  _E  o.  `' R ) )  <->  E. y  e.  x  A. z  e.  x  -.  z R y )
247, 23imbi12i 316 . . 3  |-  ( ( x  e.  ( ~P A  \  { (/) } )  ->  x  e.  ran  (  _E  \  (  _E  o.  `' R ) ) )  <->  ( (
x  C_  A  /\  x  =/=  (/) )  ->  E. y  e.  x  A. z  e.  x  -.  z R y ) )
2524albii 1571 . 2  |-  ( A. x ( x  e.  ( ~P A  \  { (/) } )  ->  x  e.  ran  (  _E 
\  (  _E  o.  `' R ) ) )  <->  A. x ( ( x 
C_  A  /\  x  =/=  (/) )  ->  E. y  e.  x  A. z  e.  x  -.  z R y ) )
26 dfss2 3255 . 2  |-  ( ( ~P A  \  { (/)
} )  C_  ran  (  _E  \  (  _E  o.  `' R ) )  <->  A. x ( x  e.  ( ~P A  \  { (/) } )  ->  x  e.  ran  (  _E 
\  (  _E  o.  `' R ) ) ) )
27 df-fr 4455 . 2  |-  ( R  Fr  A  <->  A. x
( ( x  C_  A  /\  x  =/=  (/) )  ->  E. y  e.  x  A. z  e.  x  -.  z R y ) )
2825, 26, 273bitr4ri 269 1  |-  ( R  Fr  A  <->  ( ~P A  \  { (/) } ) 
C_  ran  (  _E  \  (  _E  o.  `' R ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358   A.wal 1545   E.wex 1546    e. wcel 1715    =/= wne 2529   A.wral 2628   E.wrex 2629    \ cdif 3235    C_ wss 3238   (/)c0 3543   ~Pcpw 3714   {csn 3729   class class class wbr 4125    _E cep 4406    Fr wfr 4452   `'ccnv 4791   ran crn 4793    o. ccom 4796
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pr 4316
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-rab 2637  df-v 2875  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-op 3738  df-br 4126  df-opab 4180  df-eprel 4408  df-fr 4455  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803
  Copyright terms: Public domain W3C validator