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Theorem dffr4 25206
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 5176 . 2  |-  ( R  Fr  A  <->  A. x
( ( x  C_  A  /\  x  =/=  (/) )  ->  E. y  e.  x  ( x  i^i  ( `' R " { y } ) )  =  (/) ) )
2 df-pred 25192 . . . . . 6  |-  Pred ( R ,  x , 
y )  =  ( x  i^i  ( `' R " { y } ) )
32eqeq1i 2394 . . . . 5  |-  ( Pred ( R ,  x ,  y )  =  (/) 
<->  ( x  i^i  ( `' R " { y } ) )  =  (/) )
43rexbii 2674 . . . 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 1572 . 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 1546    = wceq 1649    =/= wne 2550   E.wrex 2650    i^i cin 3262    C_ wss 3263   (/)c0 3571   {csn 3757    Fr wfr 4479   `'ccnv 4817   "cima 4821   Predcpred 25191
This theorem is referenced by:  frmin  25266
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 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pr 4344
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 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-br 4154  df-opab 4208  df-fr 4482  df-xp 4824  df-cnv 4826  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-pred 25192
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