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Theorem tfr2ALT 8012
Description: Alternate proof of tfr2 8009 using well-founded recursion. (Contributed by Scott Fenton, 3-Aug-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
Hypothesis
Ref Expression
tfrALT.1 𝐹 = recs(𝐺)
Assertion
Ref Expression
tfr2ALT (𝐴 ∈ On → (𝐹𝐴) = (𝐺‘(𝐹𝐴)))

Proof of Theorem tfr2ALT
StepHypRef Expression
1 epweon 7472 . . 3 E We On
2 epse 5511 . . 3 E Se On
3 tfrALT.1 . . . 4 𝐹 = recs(𝐺)
4 df-recs 7983 . . . 4 recs(𝐺) = wrecs( E , On, 𝐺)
53, 4eqtri 2844 . . 3 𝐹 = wrecs( E , On, 𝐺)
61, 2, 5wfr2 7949 . 2 (𝐴 ∈ On → (𝐹𝐴) = (𝐺‘(𝐹 ↾ Pred( E , On, 𝐴))))
7 predon 7481 . . . 4 (𝐴 ∈ On → Pred( E , On, 𝐴) = 𝐴)
87reseq2d 5826 . . 3 (𝐴 ∈ On → (𝐹 ↾ Pred( E , On, 𝐴)) = (𝐹𝐴))
98fveq2d 6647 . 2 (𝐴 ∈ On → (𝐺‘(𝐹 ↾ Pred( E , On, 𝐴))) = (𝐺‘(𝐹𝐴)))
106, 9eqtrd 2856 1 (𝐴 ∈ On → (𝐹𝐴) = (𝐺‘(𝐹𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2115   E cep 5437  cres 5530  Predcpred 6120  Oncon0 6164  cfv 6328  wrecscwrecs 7921  recscrecs 7982
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-rep 5163  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-ral 3131  df-rex 3132  df-reu 3133  df-rmo 3134  df-rab 3135  df-v 3473  df-sbc 3750  df-csb 3858  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4267  df-if 4441  df-sn 4541  df-pr 4543  df-tp 4545  df-op 4547  df-uni 4812  df-iun 4894  df-br 5040  df-opab 5102  df-mpt 5120  df-tr 5146  df-id 5433  df-eprel 5438  df-po 5447  df-so 5448  df-fr 5487  df-se 5488  df-we 5489  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-pred 6121  df-ord 6167  df-on 6168  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-wrecs 7922  df-recs 7983
This theorem is referenced by: (None)
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