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Theorem dffr3 5195
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 4507 . 2  |-  ( R  Fr  A  <->  A. x
( ( x  C_  A  /\  x  =/=  (/) )  ->  E. y  e.  x  { z  e.  x  |  z R y }  =  (/) ) )
2 vex 2919 . . . . . . . . 9  |-  y  e. 
_V
3 iniseg 5194 . . . . . . . . 9  |-  ( y  e.  _V  ->  ( `' R " { y } )  =  {
z  |  z R y } )
42, 3ax-mp 8 . . . . . . . 8  |-  ( `' R " { y } )  =  {
z  |  z R y }
54ineq2i 3499 . . . . . . 7  |-  ( x  i^i  ( `' R " { y } ) )  =  ( x  i^i  { z  |  z R y } )
6 dfrab3 3577 . . . . . . 7  |-  { z  e.  x  |  z R y }  =  ( x  i^i  { z  |  z R y } )
75, 6eqtr4i 2427 . . . . . 6  |-  ( x  i^i  ( `' R " { y } ) )  =  { z  e.  x  |  z R y }
87eqeq1i 2411 . . . . 5  |-  ( ( x  i^i  ( `' R " { y } ) )  =  (/) 
<->  { z  e.  x  |  z R y }  =  (/) )
98rexbii 2691 . . . 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 1572 . 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 1546    = wceq 1649    e. wcel 1721   {cab 2390    =/= wne 2567   E.wrex 2667   {crab 2670   _Vcvv 2916    i^i cin 3279    C_ wss 3280   (/)c0 3588   {csn 3774   class class class wbr 4172    Fr wfr 4498   `'ccnv 4836   "cima 4840
This theorem is referenced by:  isofrlem  6019  dffr4  25396
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 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-br 4173  df-opab 4227  df-fr 4501  df-xp 4843  df-cnv 4845  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850
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