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Theorem tfrALTlem 25559
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 4766 . . . 4  |-  _E  We  On
2 epse 4567 . . . 4  |-  _E Se  On
3 eqid 2438 . . . 4  |- wrecs (  _E  ,  On ,  G
)  = wrecs (  _E  ,  On ,  G )
41, 2, 3wfr1 25556 . . 3  |- wrecs (  _E  ,  On ,  G
)  Fn  On
51, 2, 3wfr2 25557 . . . . 5  |-  ( y  e.  On  ->  (wrecs (  _E  ,  On ,  G ) `  y
)  =  ( G `
 (wrecs (  _E  ,  On ,  G
)  |`  Pred (  _E  ,  On ,  y )
) ) )
6 predon 25470 . . . . . . 7  |-  ( y  e.  On  ->  Pred (  _E  ,  On ,  y )  =  y )
76reseq2d 5148 . . . . . 6  |-  ( y  e.  On  ->  (wrecs (  _E  ,  On ,  G )  |`  Pred (  _E  ,  On ,  y ) )  =  (wrecs (  _E  ,  On ,  G )  |`  y
) )
87fveq2d 5734 . . . . 5  |-  ( y  e.  On  ->  ( G `  (wrecs (  _E  ,  On ,  G
)  |`  Pred (  _E  ,  On ,  y )
) )  =  ( G `  (wrecs (  _E  ,  On ,  G
)  |`  y ) ) )
95, 8eqtrd 2470 . . . 4  |-  ( y  e.  On  ->  (wrecs (  _E  ,  On ,  G ) `  y
)  =  ( G `
 (wrecs (  _E  ,  On ,  G
)  |`  y ) ) )
109rgen 2773 . . 3  |-  A. y  e.  On  (wrecs (  _E  ,  On ,  G
) `  y )  =  ( G `  (wrecs (  _E  ,  On ,  G )  |`  y
) )
11 eqid 2438 . . . 4  |- recs ( G )  = recs ( G )
1211tfr3 6662 . . 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 655 . 2  |- wrecs (  _E  ,  On ,  G
)  = recs ( G )
1413eqcomi 2442 1  |- recs ( G )  = wrecs (  _E  ,  On ,  G
)
Colors of variables: wff set class
Syntax hints:    = wceq 1653    e. wcel 1726   A.wral 2707    _E cep 4494   Oncon0 4583    |` cres 4882    Fn wfn 5451   ` cfv 5456  recscrecs 6634   Predcpred 25440  wrecscwrecs 25532
This theorem is referenced by:  tfr1ALT  25560  tfr2ALT  25561  tfr3ALT  25562
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-se 4544  df-we 4545  df-ord 4586  df-on 4587  df-suc 4589  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-recs 6635  df-pred 25441  df-wrecs 25533
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