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Theorem dffr3 5148
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 4461 . 2  |-  ( R  Fr  A  <->  A. x
( ( x  C_  A  /\  x  =/=  (/) )  ->  E. y  e.  x  { z  e.  x  |  z R y }  =  (/) ) )
2 vex 2876 . . . . . . . . 9  |-  y  e. 
_V
3 iniseg 5147 . . . . . . . . 9  |-  ( y  e.  _V  ->  ( `' R " { y } )  =  {
z  |  z R y } )
42, 3ax-mp 8 . . . . . . . 8  |-  ( `' R " { y } )  =  {
z  |  z R y }
54ineq2i 3455 . . . . . . 7  |-  ( x  i^i  ( `' R " { y } ) )  =  ( x  i^i  { z  |  z R y } )
6 dfrab3 3532 . . . . . . 7  |-  { z  e.  x  |  z R y }  =  ( x  i^i  { z  |  z R y } )
75, 6eqtr4i 2389 . . . . . 6  |-  ( x  i^i  ( `' R " { y } ) )  =  { z  e.  x  |  z R y }
87eqeq1i 2373 . . . . 5  |-  ( ( x  i^i  ( `' R " { y } ) )  =  (/) 
<->  { z  e.  x  |  z R y }  =  (/) )
98rexbii 2653 . . . 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 1571 . 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 1545    = wceq 1647    e. wcel 1715   {cab 2352    =/= wne 2529   E.wrex 2629   {crab 2632   _Vcvv 2873    i^i cin 3237    C_ wss 3238   (/)c0 3543   {csn 3729   class class class wbr 4125    Fr wfr 4452   `'ccnv 4791   "cima 4795
This theorem is referenced by:  isofrlem  5960  dffr4  24923
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pr 4316
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-rab 2637  df-v 2875  df-sbc 3078  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-sn 3735  df-pr 3736  df-op 3738  df-br 4126  df-opab 4180  df-fr 4455  df-xp 4798  df-cnv 4800  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805
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