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Theorem dffr4 25449
Description: Alternate definition of well-founded relation. (Contributed by Scott Fenton, 2-Feb-2011.)
Assertion
Ref Expression
dffr4  |-  ( R  Fr  A  <->  A. x
( ( x  C_  A  /\  x  =/=  (/) )  ->  E. y  e.  x  Pred ( R ,  x ,  y )  =  (/) ) )
Distinct variable groups:    x, y, A    x, R, y

Proof of Theorem dffr4
StepHypRef Expression
1 dffr3 5228 . 2  |-  ( R  Fr  A  <->  A. x
( ( x  C_  A  /\  x  =/=  (/) )  ->  E. y  e.  x  ( x  i^i  ( `' R " { y } ) )  =  (/) ) )
2 df-pred 25431 . . . . . 6  |-  Pred ( R ,  x , 
y )  =  ( x  i^i  ( `' R " { y } ) )
32eqeq1i 2442 . . . . 5  |-  ( Pred ( R ,  x ,  y )  =  (/) 
<->  ( x  i^i  ( `' R " { y } ) )  =  (/) )
43rexbii 2722 . . . 4  |-  ( E. y  e.  x  Pred ( R ,  x ,  y )  =  (/)  <->  E. y  e.  x  (
x  i^i  ( `' R " { y } ) )  =  (/) )
54imbi2i 304 . . 3  |-  ( ( ( x  C_  A  /\  x  =/=  (/) )  ->  E. y  e.  x  Pred ( R ,  x ,  y )  =  (/) )  <->  ( ( x 
C_  A  /\  x  =/=  (/) )  ->  E. y  e.  x  ( x  i^i  ( `' R " { y } ) )  =  (/) ) )
65albii 1575 . 2  |-  ( A. x ( ( x 
C_  A  /\  x  =/=  (/) )  ->  E. y  e.  x  Pred ( R ,  x ,  y )  =  (/) )  <->  A. x
( ( x  C_  A  /\  x  =/=  (/) )  ->  E. y  e.  x  ( x  i^i  ( `' R " { y } ) )  =  (/) ) )
71, 6bitr4i 244 1  |-  ( R  Fr  A  <->  A. x
( ( x  C_  A  /\  x  =/=  (/) )  ->  E. y  e.  x  Pred ( R ,  x ,  y )  =  (/) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   A.wal 1549    = wceq 1652    =/= wne 2598   E.wrex 2698    i^i cin 3311    C_ wss 3312   (/)c0 3620   {csn 3806    Fr wfr 4530   `'ccnv 4869   "cima 4873   Predcpred 25430
This theorem is referenced by:  frmin  25509
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  df-pred 25431
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