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Theorem wfrlem8 6998
Description: Lemma for well-founded recursion. Compute the prececessor class for an  R minimal element of  ( A  \  dom  F ). (Contributed by Scott Fenton, 21-Apr-2011.)
Hypothesis
Ref Expression
wfrlem6.1  |-  F  = wrecs ( R ,  A ,  G )
Assertion
Ref Expression
wfrlem8  |-  ( Pred ( R ,  ( A  \  dom  F
) ,  X )  =  (/)  <->  Pred ( R ,  A ,  X )  =  Pred ( R ,  dom  F ,  X ) )

Proof of Theorem wfrlem8
StepHypRef Expression
1 wfrlem6.1 . . . . 5  |-  F  = wrecs ( R ,  A ,  G )
21wfrdmss 6997 . . . 4  |-  dom  F  C_  A
3 predpredss 5348 . . . 4  |-  ( dom 
F  C_  A  ->  Pred ( R ,  dom  F ,  X )  C_  Pred ( R ,  A ,  X ) )
42, 3ax-mp 5 . . 3  |-  Pred ( R ,  dom  F ,  X )  C_  Pred ( R ,  A ,  X )
54biantru 507 . 2  |-  ( Pred ( R ,  A ,  X )  C_  Pred ( R ,  dom  F ,  X )  <->  ( Pred ( R ,  A ,  X )  C_  Pred ( R ,  dom  F ,  X )  /\  Pred ( R ,  dom  F ,  X )  C_  Pred ( R ,  A ,  X ) ) )
6 preddif 5367 . . . 4  |-  Pred ( R ,  ( A  \  dom  F ) ,  X )  =  (
Pred ( R ,  A ,  X )  \  Pred ( R ,  dom  F ,  X ) )
76eqeq1i 2433 . . 3  |-  ( Pred ( R ,  ( A  \  dom  F
) ,  X )  =  (/)  <->  ( Pred ( R ,  A ,  X )  \  Pred ( R ,  dom  F ,  X ) )  =  (/) )
8 ssdif0 3796 . . 3  |-  ( Pred ( R ,  A ,  X )  C_  Pred ( R ,  dom  F ,  X )  <->  ( Pred ( R ,  A ,  X )  \  Pred ( R ,  dom  F ,  X ) )  =  (/) )
97, 8bitr4i 255 . 2  |-  ( Pred ( R ,  ( A  \  dom  F
) ,  X )  =  (/)  <->  Pred ( R ,  A ,  X )  C_ 
Pred ( R ,  dom  F ,  X ) )
10 eqss 3422 . 2  |-  ( Pred ( R ,  A ,  X )  =  Pred ( R ,  dom  F ,  X )  <->  ( Pred ( R ,  A ,  X )  C_  Pred ( R ,  dom  F ,  X )  /\  Pred ( R ,  dom  F ,  X )  C_  Pred ( R ,  A ,  X ) ) )
115, 9, 103bitr4i 280 1  |-  ( Pred ( R ,  ( A  \  dom  F
) ,  X )  =  (/)  <->  Pred ( R ,  A ,  X )  =  Pred ( R ,  dom  F ,  X ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 187    /\ wa 370    = wceq 1437    \ cdif 3376    C_ wss 3379   (/)c0 3704   dom cdm 4796   Predcpred 5341  wrecscwrecs 6982
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-ral 2719  df-rex 2720  df-rab 2723  df-v 3024  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-nul 3705  df-if 3855  df-sn 3942  df-pr 3944  df-op 3948  df-uni 4163  df-iun 4244  df-br 4367  df-opab 4426  df-xp 4802  df-rel 4803  df-cnv 4804  df-co 4805  df-dm 4806  df-rn 4807  df-res 4808  df-ima 4809  df-pred 5342  df-iota 5508  df-fun 5546  df-fn 5547  df-fv 5552  df-wrecs 6983
This theorem is referenced by:  wfrlem10  7000
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