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Theorem tfr2ALT 27882
Description: tfr2 6960 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 6498 . . 3  |-  _E  We  On
2 epse 4804 . . 3  |-  _E Se  On
3 tfrALT.1 . . . 4  |-  F  = recs ( G )
4 tfrALTlem 27880 . . . 4  |- recs ( G )  = wrecs (  _E  ,  On ,  G
)
53, 4eqtri 2480 . . 3  |-  F  = wrecs (  _E  ,  On ,  G )
61, 2, 5wfr2 27878 . 2  |-  ( z  e.  On  ->  ( F `  z )  =  ( G `  ( F  |`  Pred (  _E  ,  On ,  z ) ) ) )
7 predon 27791 . . . 4  |-  ( z  e.  On  ->  Pred (  _E  ,  On ,  z )  =  z )
87reseq2d 5211 . . 3  |-  ( z  e.  On  ->  ( F  |`  Pred (  _E  ,  On ,  z )
)  =  ( F  |`  z ) )
98fveq2d 5796 . 2  |-  ( z  e.  On  ->  ( G `  ( F  |` 
Pred (  _E  ,  On ,  z )
) )  =  ( G `  ( F  |`  z ) ) )
106, 9eqtrd 2492 1  |-  ( z  e.  On  ->  ( F `  z )  =  ( G `  ( F  |`  z ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370    e. wcel 1758    _E cep 4731   Oncon0 4820    |` cres 4943   ` cfv 5519  recscrecs 6934   Predcpred 27761  wrecscwrecs 27853
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-pss 3445  df-nul 3739  df-if 3893  df-sn 3979  df-pr 3981  df-tp 3983  df-op 3985  df-uni 4193  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-tr 4487  df-eprel 4733  df-id 4737  df-po 4742  df-so 4743  df-fr 4780  df-se 4781  df-we 4782  df-ord 4823  df-on 4824  df-suc 4826  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-recs 6935  df-pred 27762  df-wrecs 27854
This theorem is referenced by: (None)
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