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Theorem tfr2ALT 29332
Description: tfr2 7065 via well-founded recursion. (Contributed by Scott Fenton, 22-Apr-2011.) (Revised by Mario Carneiro, 26-Jun-2015.) (New usage is discouraged.) (Proof modification is discouraged.)
Hypothesis
Ref Expression
tfrALT.1  |-  F  = recs ( G )
Assertion
Ref Expression
tfr2ALT  |-  ( z  e.  On  ->  ( F `  z )  =  ( G `  ( F  |`  z ) ) )

Proof of Theorem tfr2ALT
StepHypRef Expression
1 epweon 6600 . . 3  |-  _E  We  On
2 epse 4848 . . 3  |-  _E Se  On
3 tfrALT.1 . . . 4  |-  F  = recs ( G )
4 tfrALTlem 29330 . . . 4  |- recs ( G )  = wrecs (  _E  ,  On ,  G
)
53, 4eqtri 2470 . . 3  |-  F  = wrecs (  _E  ,  On ,  G )
61, 2, 5wfr2 29328 . 2  |-  ( z  e.  On  ->  ( F `  z )  =  ( G `  ( F  |`  Pred (  _E  ,  On ,  z ) ) ) )
7 predon 29241 . . . 4  |-  ( z  e.  On  ->  Pred (  _E  ,  On ,  z )  =  z )
87reseq2d 5259 . . 3  |-  ( z  e.  On  ->  ( F  |`  Pred (  _E  ,  On ,  z )
)  =  ( F  |`  z ) )
98fveq2d 5856 . 2  |-  ( z  e.  On  ->  ( G `  ( F  |` 
Pred (  _E  ,  On ,  z )
) )  =  ( G `  ( F  |`  z ) ) )
106, 9eqtrd 2482 1  |-  ( z  e.  On  ->  ( F `  z )  =  ( G `  ( F  |`  z ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1381    e. wcel 1802    _E cep 4775   Oncon0 4864    |` cres 4987   ` cfv 5574  recscrecs 7039   Predcpred 29211  wrecscwrecs 29303
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-pss 3474  df-nul 3768  df-if 3923  df-sn 4011  df-pr 4013  df-tp 4015  df-op 4017  df-uni 4231  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-tr 4527  df-eprel 4777  df-id 4781  df-po 4786  df-so 4787  df-fr 4824  df-se 4825  df-we 4826  df-ord 4867  df-on 4868  df-suc 4870  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-recs 7040  df-pred 29212  df-wrecs 29304
This theorem is referenced by: (None)
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