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Theorem fri 4485
Description: Property of well-founded relation (one direction of definition). (Contributed by NM, 18-Mar-1997.)
Assertion
Ref Expression
fri  |-  ( ( ( B  e.  C  /\  R  Fr  A
)  /\  ( B  C_  A  /\  B  =/=  (/) ) )  ->  E. x  e.  B  A. y  e.  B  -.  y R x )
Distinct variable groups:    x, y, A    x, B, y    x, R, y
Allowed substitution hints:    C( x, y)

Proof of Theorem fri
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 df-fr 4482 . . 3  |-  ( R  Fr  A  <->  A. z
( ( z  C_  A  /\  z  =/=  (/) )  ->  E. x  e.  z  A. y  e.  z  -.  y R x ) )
2 sseq1 3312 . . . . . 6  |-  ( z  =  B  ->  (
z  C_  A  <->  B  C_  A
) )
3 neeq1 2558 . . . . . 6  |-  ( z  =  B  ->  (
z  =/=  (/)  <->  B  =/=  (/) ) )
42, 3anbi12d 692 . . . . 5  |-  ( z  =  B  ->  (
( z  C_  A  /\  z  =/=  (/) )  <->  ( B  C_  A  /\  B  =/=  (/) ) ) )
5 raleq 2847 . . . . . 6  |-  ( z  =  B  ->  ( A. y  e.  z  -.  y R x  <->  A. y  e.  B  -.  y R x ) )
65rexeqbi1dv 2856 . . . . 5  |-  ( z  =  B  ->  ( E. x  e.  z  A. y  e.  z  -.  y R x  <->  E. x  e.  B  A. y  e.  B  -.  y R x ) )
74, 6imbi12d 312 . . . 4  |-  ( z  =  B  ->  (
( ( z  C_  A  /\  z  =/=  (/) )  ->  E. x  e.  z  A. y  e.  z  -.  y R x )  <-> 
( ( B  C_  A  /\  B  =/=  (/) )  ->  E. x  e.  B  A. y  e.  B  -.  y R x ) ) )
87spcgv 2979 . . 3  |-  ( B  e.  C  ->  ( A. z ( ( z 
C_  A  /\  z  =/=  (/) )  ->  E. x  e.  z  A. y  e.  z  -.  y R x )  -> 
( ( B  C_  A  /\  B  =/=  (/) )  ->  E. x  e.  B  A. y  e.  B  -.  y R x ) ) )
91, 8syl5bi 209 . 2  |-  ( B  e.  C  ->  ( R  Fr  A  ->  ( ( B  C_  A  /\  B  =/=  (/) )  ->  E. x  e.  B  A. y  e.  B  -.  y R x ) ) )
109imp31 422 1  |-  ( ( ( B  e.  C  /\  R  Fr  A
)  /\  ( B  C_  A  /\  B  =/=  (/) ) )  ->  E. x  e.  B  A. y  e.  B  -.  y R x )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359   A.wal 1546    = wceq 1649    e. wcel 1717    =/= wne 2550   A.wral 2649   E.wrex 2650    C_ wss 3263   (/)c0 3571   class class class wbr 4153    Fr wfr 4479
This theorem is referenced by:  frc  4489  fr2nr  4501  frminex  4503  wereu  4519  wereu2  4520  fr3nr  4700  frfi  7288  fimax2g  7289  wofib  7447  wemapso  7453  wemapso2  7454  noinfep  7547  noinfepOLD  7548  cflim2  8076  isfin1-3  8199  fin12  8226  fpwwe2lem12  8449  fpwwe2lem13  8450  fpwwe2  8451  frinfm  26128  fdc  26140  fnwe2lem2  26817  bnj110  28567
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 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-v 2901  df-in 3270  df-ss 3277  df-fr 4482
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