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Theorem dffr3 4952
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
StepHypRef Expression
1 dffr2 4251 . 2  |-  ( R  Fr  A  <->  A. x
( ( x  C_  A  /\  x  =/=  (/) )  ->  E. y  e.  x  { z  e.  x  |  z R y }  =  (/) ) )
2 vex 2730 . . . . . . . . 9  |-  y  e. 
_V
3 iniseg 4951 . . . . . . . . 9  |-  ( y  e.  _V  ->  ( `' R " { y } )  =  {
z  |  z R y } )
42, 3ax-mp 10 . . . . . . . 8  |-  ( `' R " { y } )  =  {
z  |  z R y }
54ineq2i 3275 . . . . . . 7  |-  ( x  i^i  ( `' R " { y } ) )  =  ( x  i^i  { z  |  z R y } )
6 dfrab3 3351 . . . . . . 7  |-  { z  e.  x  |  z R y }  =  ( x  i^i  { z  |  z R y } )
75, 6eqtr4i 2276 . . . . . 6  |-  ( x  i^i  ( `' R " { y } ) )  =  { z  e.  x  |  z R y }
87eqeq1i 2260 . . . . 5  |-  ( ( x  i^i  ( `' R " { y } ) )  =  (/) 
<->  { z  e.  x  |  z R y }  =  (/) )
98rexbii 2532 . . . 4  |-  ( E. y  e.  x  ( x  i^i  ( `' R " { y } ) )  =  (/) 
<->  E. y  e.  x  { z  e.  x  |  z R y }  =  (/) )
109imbi2i 305 . . 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 1554 . 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 245 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 6    <-> wb 178    /\ wa 360   A.wal 1532    = wceq 1619    e. wcel 1621   {cab 2239    =/= wne 2412   E.wrex 2510   {crab 2512   _Vcvv 2727    i^i cin 3077    C_ wss 3078   (/)c0 3362   {csn 3544   class class class wbr 3920    Fr wfr 4242   `'ccnv 4579   "cima 4583
This theorem is referenced by:  isofrlem  5689  dffr4  23350
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pr 4108
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-rab 2516  df-v 2729  df-sbc 2922  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-sn 3550  df-pr 3551  df-op 3553  df-br 3921  df-opab 3975  df-fr 4245  df-xp 4594  df-cnv 4596  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601
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