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Theorem dffr3 5044
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 4357 . 2  |-  ( R  Fr  A  <->  A. x
( ( x  C_  A  /\  x  =/=  (/) )  ->  E. y  e.  x  { z  e.  x  |  z R y }  =  (/) ) )
2 vex 2792 . . . . . . . . 9  |-  y  e. 
_V
3 iniseg 5043 . . . . . . . . 9  |-  ( y  e.  _V  ->  ( `' R " { y } )  =  {
z  |  z R y } )
42, 3ax-mp 8 . . . . . . . 8  |-  ( `' R " { y } )  =  {
z  |  z R y }
54ineq2i 3368 . . . . . . 7  |-  ( x  i^i  ( `' R " { y } ) )  =  ( x  i^i  { z  |  z R y } )
6 dfrab3 3445 . . . . . . 7  |-  { z  e.  x  |  z R y }  =  ( x  i^i  { z  |  z R y } )
75, 6eqtr4i 2307 . . . . . 6  |-  ( x  i^i  ( `' R " { y } ) )  =  { z  e.  x  |  z R y }
87eqeq1i 2291 . . . . 5  |-  ( ( x  i^i  ( `' R " { y } ) )  =  (/) 
<->  { z  e.  x  |  z R y }  =  (/) )
98rexbii 2569 . . . 4  |-  ( E. y  e.  x  ( x  i^i  ( `' R " { y } ) )  =  (/) 
<->  E. y  e.  x  { z  e.  x  |  z R y }  =  (/) )
109imbi2i 303 . . 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 1553 . 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 243 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 176    /\ wa 358   A.wal 1527    = wceq 1623    e. wcel 1685   {cab 2270    =/= wne 2447   E.wrex 2545   {crab 2548   _Vcvv 2789    i^i cin 3152    C_ wss 3153   (/)c0 3456   {csn 3641   class class class wbr 4024    Fr wfr 4348   `'ccnv 4687   "cima 4691
This theorem is referenced by:  isofrlem  5799  dffr4  23586
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 1636  ax-8 1644  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pr 4213
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 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-sbc 2993  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-sn 3647  df-pr 3648  df-op 3650  df-br 4025  df-opab 4079  df-fr 4351  df-xp 4694  df-cnv 4696  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701
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