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Theorem fri 4785
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 4782 . . 3  |-  ( R  Fr  A  <->  A. z
( ( z  C_  A  /\  z  =/=  (/) )  ->  E. x  e.  z  A. y  e.  z  -.  y R x ) )
2 sseq1 3463 . . . . . 6  |-  ( z  =  B  ->  (
z  C_  A  <->  B  C_  A
) )
3 neeq1 2684 . . . . . 6  |-  ( z  =  B  ->  (
z  =/=  (/)  <->  B  =/=  (/) ) )
42, 3anbi12d 709 . . . . 5  |-  ( z  =  B  ->  (
( z  C_  A  /\  z  =/=  (/) )  <->  ( B  C_  A  /\  B  =/=  (/) ) ) )
5 raleq 3004 . . . . . 6  |-  ( z  =  B  ->  ( A. y  e.  z  -.  y R x  <->  A. y  e.  B  -.  y R x ) )
65rexeqbi1dv 3013 . . . . 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 318 . . . 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 3144 . . 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 217 . 2  |-  ( B  e.  C  ->  ( R  Fr  A  ->  ( ( B  C_  A  /\  B  =/=  (/) )  ->  E. x  e.  B  A. y  e.  B  -.  y R x ) ) )
109imp31 430 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 367   A.wal 1403    = wceq 1405    e. wcel 1842    =/= wne 2598   A.wral 2754   E.wrex 2755    C_ wss 3414   (/)c0 3738   class class class wbr 4395    Fr wfr 4779
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380
This theorem depends on definitions:  df-bi 185  df-an 369  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-v 3061  df-in 3421  df-ss 3428  df-fr 4782
This theorem is referenced by:  frc  4789  fr2nr  4801  frminex  4803  wereu  4819  wereu2  4820  fr3nr  6597  frfi  7799  fimax2g  7800  wofib  8004  wemapso  8010  wemapso2OLD  8011  wemapso2lem  8012  noinfep  8109  cflim2  8675  isfin1-3  8798  fin12  8825  fpwwe2lem12  9049  fpwwe2lem13  9050  fpwwe2  9051  bnj110  29243  frinfm  31508  fdc  31520  fnwe2lem2  35359
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