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Theorem dffr4 24182
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 5045 . 2  |-  ( R  Fr  A  <->  A. x
( ( x  C_  A  /\  x  =/=  (/) )  ->  E. y  e.  x  ( x  i^i  ( `' R " { y } ) )  =  (/) ) )
2 df-pred 24168 . . . . . 6  |-  Pred ( R ,  x , 
y )  =  ( x  i^i  ( `' R " { y } ) )
32eqeq1i 2290 . . . . 5  |-  ( Pred ( R ,  x ,  y )  =  (/) 
<->  ( x  i^i  ( `' R " { y } ) )  =  (/) )
43rexbii 2568 . . . 4  |-  ( E. y  e.  x  Pred ( R ,  x ,  y )  =  (/)  <->  E. y  e.  x  (
x  i^i  ( `' R " { y } ) )  =  (/) )
54imbi2i 303 . . 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 1553 . 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 243 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 176    /\ wa 358   A.wal 1527    = wceq 1623    =/= wne 2446   E.wrex 2544    i^i cin 3151    C_ wss 3152   (/)c0 3455   {csn 3640    Fr wfr 4349   `'ccnv 4688   "cima 4692   Predcpred 24167
This theorem is referenced by:  frmin  24242
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-fr 4352  df-xp 4695  df-cnv 4697  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-pred 24168
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