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Theorem wfrlem8 29515
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 )
21wfrlem7 29514 . . . 4  |-  dom  F  C_  A
3 predpredss 29415 . . . 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 503 . 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 29436 . . . 4  |-  Pred ( R ,  ( A  \  dom  F ) ,  X )  =  (
Pred ( R ,  A ,  X )  \  Pred ( R ,  dom  F ,  X ) )
76eqeq1i 2389 . . 3  |-  ( Pred ( R ,  ( A  \  dom  F
) ,  X )  =  (/)  <->  ( Pred ( R ,  A ,  X )  \  Pred ( R ,  dom  F ,  X ) )  =  (/) )
8 ssdif0 3801 . . 3  |-  ( Pred ( R ,  A ,  X )  C_  Pred ( R ,  dom  F ,  X )  <->  ( Pred ( R ,  A ,  X )  \  Pred ( R ,  dom  F ,  X ) )  =  (/) )
97, 8bitr4i 252 . 2  |-  ( Pred ( R ,  ( A  \  dom  F
) ,  X )  =  (/)  <->  Pred ( R ,  A ,  X )  C_ 
Pred ( R ,  dom  F ,  X ) )
10 eqss 3432 . 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 277 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 184    /\ wa 367    = wceq 1399    \ cdif 3386    C_ wss 3389   (/)c0 3711   dom cdm 4913   Predcpred 29408  wrecscwrecs 29500
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-rab 2741  df-v 3036  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-iun 4245  df-br 4368  df-opab 4426  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-fv 5504  df-pred 29409  df-wrecs 29501
This theorem is referenced by:  wfrlem10  29517
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