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Theorem tfrALTlem 28939
Description: Lemma for deriving transfinite recursion from well-founded recursion. (Contributed by Scott Fenton, 22-Apr-2011.)
Assertion
Ref Expression
tfrALTlem  |- recs ( G )  = wrecs (  _E  ,  On ,  G
)

Proof of Theorem tfrALTlem
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 epweon 6597 . . . 4  |-  _E  We  On
2 epse 4862 . . . 4  |-  _E Se  On
3 eqid 2467 . . . 4  |- wrecs (  _E  ,  On ,  G
)  = wrecs (  _E  ,  On ,  G )
41, 2, 3wfr1 28936 . . 3  |- wrecs (  _E  ,  On ,  G
)  Fn  On
51, 2, 3wfr2 28937 . . . . 5  |-  ( y  e.  On  ->  (wrecs (  _E  ,  On ,  G ) `  y
)  =  ( G `
 (wrecs (  _E  ,  On ,  G
)  |`  Pred (  _E  ,  On ,  y )
) ) )
6 predon 28850 . . . . . . 7  |-  ( y  e.  On  ->  Pred (  _E  ,  On ,  y )  =  y )
76reseq2d 5271 . . . . . 6  |-  ( y  e.  On  ->  (wrecs (  _E  ,  On ,  G )  |`  Pred (  _E  ,  On ,  y ) )  =  (wrecs (  _E  ,  On ,  G )  |`  y
) )
87fveq2d 5868 . . . . 5  |-  ( y  e.  On  ->  ( G `  (wrecs (  _E  ,  On ,  G
)  |`  Pred (  _E  ,  On ,  y )
) )  =  ( G `  (wrecs (  _E  ,  On ,  G
)  |`  y ) ) )
95, 8eqtrd 2508 . . . 4  |-  ( y  e.  On  ->  (wrecs (  _E  ,  On ,  G ) `  y
)  =  ( G `
 (wrecs (  _E  ,  On ,  G
)  |`  y ) ) )
109rgen 2824 . . 3  |-  A. y  e.  On  (wrecs (  _E  ,  On ,  G
) `  y )  =  ( G `  (wrecs (  _E  ,  On ,  G )  |`  y
) )
11 eqid 2467 . . . 4  |- recs ( G )  = recs ( G )
1211tfr3 7065 . . 3  |-  ( (wrecs (  _E  ,  On ,  G )  Fn  On  /\ 
A. y  e.  On  (wrecs (  _E  ,  On ,  G ) `  y
)  =  ( G `
 (wrecs (  _E  ,  On ,  G
)  |`  y ) ) )  -> wrecs (  _E  ,  On ,  G )  = recs ( G ) )
134, 10, 12mp2an 672 . 2  |- wrecs (  _E  ,  On ,  G
)  = recs ( G )
1413eqcomi 2480 1  |- recs ( G )  = wrecs (  _E  ,  On ,  G
)
Colors of variables: wff setvar class
Syntax hints:    = wceq 1379    e. wcel 1767   A.wral 2814    _E cep 4789   Oncon0 4878    |` cres 5001    Fn wfn 5581   ` cfv 5586  recscrecs 7038   Predcpred 28820  wrecscwrecs 28912
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-recs 7039  df-pred 28821  df-wrecs 28913
This theorem is referenced by:  tfr1ALT  28940  tfr2ALT  28941  tfr3ALT  28942
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