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Theorem dffr3 5228
Description: Alternate definition of well-founded relation. Definition 6.21 of [TakeutiZaring] p. 30. (Contributed by NM, 23-Apr-2004.) (Revised by Mario Carneiro, 23-Jun-2015.)
Assertion
Ref Expression
dffr3  |-  ( R  Fr  A  <->  A. x
( ( x  C_  A  /\  x  =/=  (/) )  ->  E. y  e.  x  ( x  i^i  ( `' R " { y } ) )  =  (/) ) )
Distinct variable groups:    x, y, A    x, R, y

Proof of Theorem dffr3
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 dffr2 4539 . 2  |-  ( R  Fr  A  <->  A. x
( ( x  C_  A  /\  x  =/=  (/) )  ->  E. y  e.  x  { z  e.  x  |  z R y }  =  (/) ) )
2 vex 2951 . . . . . . . . 9  |-  y  e. 
_V
3 iniseg 5227 . . . . . . . . 9  |-  ( y  e.  _V  ->  ( `' R " { y } )  =  {
z  |  z R y } )
42, 3ax-mp 8 . . . . . . . 8  |-  ( `' R " { y } )  =  {
z  |  z R y }
54ineq2i 3531 . . . . . . 7  |-  ( x  i^i  ( `' R " { y } ) )  =  ( x  i^i  { z  |  z R y } )
6 dfrab3 3609 . . . . . . 7  |-  { z  e.  x  |  z R y }  =  ( x  i^i  { z  |  z R y } )
75, 6eqtr4i 2458 . . . . . 6  |-  ( x  i^i  ( `' R " { y } ) )  =  { z  e.  x  |  z R y }
87eqeq1i 2442 . . . . 5  |-  ( ( x  i^i  ( `' R " { y } ) )  =  (/) 
<->  { z  e.  x  |  z R y }  =  (/) )
98rexbii 2722 . . . 4  |-  ( E. y  e.  x  ( x  i^i  ( `' R " { y } ) )  =  (/) 
<->  E. y  e.  x  { z  e.  x  |  z R y }  =  (/) )
109imbi2i 304 . . 3  |-  ( ( ( x  C_  A  /\  x  =/=  (/) )  ->  E. y  e.  x  ( x  i^i  ( `' R " { y } ) )  =  (/) )  <->  ( ( x 
C_  A  /\  x  =/=  (/) )  ->  E. y  e.  x  { z  e.  x  |  z R y }  =  (/) ) )
1110albii 1575 . 2  |-  ( A. x ( ( x 
C_  A  /\  x  =/=  (/) )  ->  E. y  e.  x  ( x  i^i  ( `' R " { y } ) )  =  (/) )  <->  A. x
( ( x  C_  A  /\  x  =/=  (/) )  ->  E. y  e.  x  { z  e.  x  |  z R y }  =  (/) ) )
121, 11bitr4i 244 1  |-  ( R  Fr  A  <->  A. x
( ( x  C_  A  /\  x  =/=  (/) )  ->  E. y  e.  x  ( x  i^i  ( `' R " { y } ) )  =  (/) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   A.wal 1549    = wceq 1652    e. wcel 1725   {cab 2421    =/= wne 2598   E.wrex 2698   {crab 2701   _Vcvv 2948    i^i cin 3311    C_ wss 3312   (/)c0 3620   {csn 3806   class class class wbr 4204    Fr wfr 4530   `'ccnv 4869   "cima 4873
This theorem is referenced by:  isofrlem  6052  dffr4  25449
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-opab 4259  df-fr 4533  df-xp 4876  df-cnv 4878  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883
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