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Theorem tfrALTlem 27758
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 6410 . . . 4  |-  _E  We  On
2 epse 4718 . . . 4  |-  _E Se  On
3 eqid 2443 . . . 4  |- wrecs (  _E  ,  On ,  G
)  = wrecs (  _E  ,  On ,  G )
41, 2, 3wfr1 27755 . . 3  |- wrecs (  _E  ,  On ,  G
)  Fn  On
51, 2, 3wfr2 27756 . . . . 5  |-  ( y  e.  On  ->  (wrecs (  _E  ,  On ,  G ) `  y
)  =  ( G `
 (wrecs (  _E  ,  On ,  G
)  |`  Pred (  _E  ,  On ,  y )
) ) )
6 predon 27669 . . . . . . 7  |-  ( y  e.  On  ->  Pred (  _E  ,  On ,  y )  =  y )
76reseq2d 5125 . . . . . 6  |-  ( y  e.  On  ->  (wrecs (  _E  ,  On ,  G )  |`  Pred (  _E  ,  On ,  y ) )  =  (wrecs (  _E  ,  On ,  G )  |`  y
) )
87fveq2d 5710 . . . . 5  |-  ( y  e.  On  ->  ( G `  (wrecs (  _E  ,  On ,  G
)  |`  Pred (  _E  ,  On ,  y )
) )  =  ( G `  (wrecs (  _E  ,  On ,  G
)  |`  y ) ) )
95, 8eqtrd 2475 . . . 4  |-  ( y  e.  On  ->  (wrecs (  _E  ,  On ,  G ) `  y
)  =  ( G `
 (wrecs (  _E  ,  On ,  G
)  |`  y ) ) )
109rgen 2796 . . 3  |-  A. y  e.  On  (wrecs (  _E  ,  On ,  G
) `  y )  =  ( G `  (wrecs (  _E  ,  On ,  G )  |`  y
) )
11 eqid 2443 . . . 4  |- recs ( G )  = recs ( G )
1211tfr3 6873 . . 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 2447 1  |- recs ( G )  = wrecs (  _E  ,  On ,  G
)
Colors of variables: wff setvar class
Syntax hints:    = wceq 1369    e. wcel 1756   A.wral 2730    _E cep 4645   Oncon0 4734    |` cres 4857    Fn wfn 5428   ` cfv 5433  recscrecs 6846   Predcpred 27639  wrecscwrecs 27731
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4418  ax-sep 4428  ax-nul 4436  ax-pow 4485  ax-pr 4546  ax-un 6387
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2735  df-rex 2736  df-reu 2737  df-rmo 2738  df-rab 2739  df-v 2989  df-sbc 3202  df-csb 3304  df-dif 3346  df-un 3348  df-in 3350  df-ss 3357  df-pss 3359  df-nul 3653  df-if 3807  df-sn 3893  df-pr 3895  df-tp 3897  df-op 3899  df-uni 4107  df-iun 4188  df-br 4308  df-opab 4366  df-mpt 4367  df-tr 4401  df-eprel 4647  df-id 4651  df-po 4656  df-so 4657  df-fr 4694  df-se 4695  df-we 4696  df-ord 4737  df-on 4738  df-suc 4740  df-xp 4861  df-rel 4862  df-cnv 4863  df-co 4864  df-dm 4865  df-rn 4866  df-res 4867  df-ima 4868  df-iota 5396  df-fun 5435  df-fn 5436  df-f 5437  df-f1 5438  df-fo 5439  df-f1o 5440  df-fv 5441  df-recs 6847  df-pred 27640  df-wrecs 27732
This theorem is referenced by:  tfr1ALT  27759  tfr2ALT  27760  tfr3ALT  27761
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