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Theorem tfr2ALT 25564
Description: tfr2 6662 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 4767 . . 3  |-  _E  We  On
2 epse 4568 . . 3  |-  _E Se  On
3 tfrALT.1 . . . 4  |-  F  = recs ( G )
4 tfrALTlem 25562 . . . 4  |- recs ( G )  = wrecs (  _E  ,  On ,  G
)
53, 4eqtri 2458 . . 3  |-  F  = wrecs (  _E  ,  On ,  G )
61, 2, 5wfr2 25560 . 2  |-  ( z  e.  On  ->  ( F `  z )  =  ( G `  ( F  |`  Pred (  _E  ,  On ,  z ) ) ) )
7 predon 25473 . . . 4  |-  ( z  e.  On  ->  Pred (  _E  ,  On ,  z )  =  z )
87reseq2d 5149 . . 3  |-  ( z  e.  On  ->  ( F  |`  Pred (  _E  ,  On ,  z )
)  =  ( F  |`  z ) )
98fveq2d 5735 . 2  |-  ( z  e.  On  ->  ( G `  ( F  |` 
Pred (  _E  ,  On ,  z )
) )  =  ( G `  ( F  |`  z ) ) )
106, 9eqtrd 2470 1  |-  ( z  e.  On  ->  ( F `  z )  =  ( G `  ( F  |`  z ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653    e. wcel 1726    _E cep 4495   Oncon0 4584    |` cres 4883   ` cfv 5457  recscrecs 6635   Predcpred 25443  wrecscwrecs 25535
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
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 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-se 4545  df-we 4546  df-ord 4587  df-on 4588  df-suc 4590  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-recs 6636  df-pred 25444  df-wrecs 25536
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