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Theorem tfr2ALT 28791
Description: tfr2 7057 via well-founded recursion. (Contributed by Scott Fenton, 22-Apr-2011.) (Revised by Mario Carneiro, 26-Jun-2015.) (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 6590 . . 3  |-  _E  We  On
2 epse 4855 . . 3  |-  _E Se  On
3 tfrALT.1 . . . 4  |-  F  = recs ( G )
4 tfrALTlem 28789 . . . 4  |- recs ( G )  = wrecs (  _E  ,  On ,  G
)
53, 4eqtri 2489 . . 3  |-  F  = wrecs (  _E  ,  On ,  G )
61, 2, 5wfr2 28787 . 2  |-  ( z  e.  On  ->  ( F `  z )  =  ( G `  ( F  |`  Pred (  _E  ,  On ,  z ) ) ) )
7 predon 28700 . . . 4  |-  ( z  e.  On  ->  Pred (  _E  ,  On ,  z )  =  z )
87reseq2d 5264 . . 3  |-  ( z  e.  On  ->  ( F  |`  Pred (  _E  ,  On ,  z )
)  =  ( F  |`  z ) )
98fveq2d 5861 . 2  |-  ( z  e.  On  ->  ( G `  ( F  |` 
Pred (  _E  ,  On ,  z )
) )  =  ( G `  ( F  |`  z ) ) )
106, 9eqtrd 2501 1  |-  ( z  e.  On  ->  ( F `  z )  =  ( G `  ( F  |`  z ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1374    e. wcel 1762    _E cep 4782   Oncon0 4871    |` cres 4994   ` cfv 5579  recscrecs 7031   Predcpred 28670  wrecscwrecs 28762
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-se 4832  df-we 4833  df-ord 4874  df-on 4875  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-recs 7032  df-pred 28671  df-wrecs 28763
This theorem is referenced by: (None)
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