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Theorem dffr5 25332
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 3298 . . . . 5  |-  ( x  e.  ( ~P A  \  { (/) } )  <->  ( x  e.  ~P A  /\  -.  x  e.  { (/) } ) )
2 vex 2927 . . . . . . 7  |-  x  e. 
_V
32elpw 3773 . . . . . 6  |-  ( x  e.  ~P A  <->  x  C_  A
)
4 elsn 3797 . . . . . . 7  |-  ( x  e.  { (/) }  <->  x  =  (/) )
54necon3bbii 2606 . . . . . 6  |-  ( -.  x  e.  { (/) }  <-> 
x  =/=  (/) )
63, 5anbi12i 679 . . . . 5  |-  ( ( x  e.  ~P A  /\  -.  x  e.  { (/)
} )  <->  ( x  C_  A  /\  x  =/=  (/) ) )
71, 6bitri 241 . . . 4  |-  ( x  e.  ( ~P A  \  { (/) } )  <->  ( x  C_  A  /\  x  =/=  (/) ) )
8 brdif 4228 . . . . . . 7  |-  ( y (  _E  \  (  _E  o.  `' R ) ) x  <->  ( y  _E  x  /\  -.  y
(  _E  o.  `' R ) x ) )
9 epel 4465 . . . . . . . 8  |-  ( y  _E  x  <->  y  e.  x )
10 vex 2927 . . . . . . . . . . 11  |-  y  e. 
_V
1110, 2coep 25330 . . . . . . . . . 10  |-  ( y (  _E  o.  `' R ) x  <->  E. z  e.  x  y `' R z )
12 vex 2927 . . . . . . . . . . . 12  |-  z  e. 
_V
1310, 12brcnv 5022 . . . . . . . . . . 11  |-  ( y `' R z  <->  z R
y )
1413rexbii 2699 . . . . . . . . . 10  |-  ( E. z  e.  x  y `' R z  <->  E. z  e.  x  z R
y )
15 dfrex2 2687 . . . . . . . . . 10  |-  ( E. z  e.  x  z R y  <->  -.  A. z  e.  x  -.  z R y )
1611, 14, 153bitrri 264 . . . . . . . . 9  |-  ( -. 
A. z  e.  x  -.  z R y  <->  y (  _E  o.  `' R ) x )
1716con1bii 322 . . . . . . . 8  |-  ( -.  y (  _E  o.  `' R ) x  <->  A. z  e.  x  -.  z R y )
189, 17anbi12i 679 . . . . . . 7  |-  ( ( y  _E  x  /\  -.  y (  _E  o.  `' R ) x )  <-> 
( y  e.  x  /\  A. z  e.  x  -.  z R y ) )
198, 18bitri 241 . . . . . 6  |-  ( y (  _E  \  (  _E  o.  `' R ) ) x  <->  ( y  e.  x  /\  A. z  e.  x  -.  z R y ) )
2019exbii 1589 . . . . 5  |-  ( E. y  y (  _E 
\  (  _E  o.  `' R ) ) x  <->  E. y ( y  e.  x  /\  A. z  e.  x  -.  z R y ) )
212elrn 5077 . . . . 5  |-  ( x  e.  ran  (  _E 
\  (  _E  o.  `' R ) )  <->  E. y 
y (  _E  \ 
(  _E  o.  `' R ) ) x )
22 df-rex 2680 . . . . 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 269 . . . 4  |-  ( x  e.  ran  (  _E 
\  (  _E  o.  `' R ) )  <->  E. y  e.  x  A. z  e.  x  -.  z R y )
247, 23imbi12i 317 . . 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 1572 . 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 3305 . 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 4509 . 2  |-  ( R  Fr  A  <->  A. x
( ( x  C_  A  /\  x  =/=  (/) )  ->  E. y  e.  x  A. z  e.  x  -.  z R y ) )
2825, 26, 273bitr4ri 270 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 177    /\ wa 359   A.wal 1546   E.wex 1547    e. wcel 1721    =/= wne 2575   A.wral 2674   E.wrex 2675    \ cdif 3285    C_ wss 3288   (/)c0 3596   ~Pcpw 3767   {csn 3782   class class class wbr 4180    _E cep 4460    Fr wfr 4506   `'ccnv 4844   ran crn 4846    o. ccom 4849
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pr 4371
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-br 4181  df-opab 4235  df-eprel 4462  df-fr 4509  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856
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