Theorem tfr2ALT 7384

Description: Alternate proof of tfr2 7381 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 → (𝐹‘𝐴) = (𝐺‘(𝐹 ↾ 𝐴)))

Proof of Theorem tfr2ALT

Step	Hyp	Ref	Expression
1		epweon 6875	. . 3 ⊢ E We On
2		epse 5021	. . 3 ⊢ E Se On
3		tfrALT.1	. . . 4 ⊢ 𝐹 = recs(𝐺)
4		df-recs 7355	. . . 4 ⊢ recs(𝐺) = wrecs( E , On, 𝐺)
5	3, 4	eqtri 2632	. . 3 ⊢ 𝐹 = wrecs( E , On, 𝐺)
6	1, 2, 5	wfr2 7321	. 2 ⊢ (𝐴 ∈ On → (𝐹‘𝐴) = (𝐺‘(𝐹 ↾ Pred( E , On, 𝐴))))
7		predon 6883	. . . 4 ⊢ (𝐴 ∈ On → Pred( E , On, 𝐴) = 𝐴)
8	7	reseq2d 5317	. . 3 ⊢ (𝐴 ∈ On → (𝐹 ↾ Pred( E , On, 𝐴)) = (𝐹 ↾ 𝐴))
9	8	fveq2d 6107	. 2 ⊢ (𝐴 ∈ On → (𝐺‘(𝐹 ↾ Pred( E , On, 𝐴))) = (𝐺‘(𝐹 ↾ 𝐴)))
10	6, 9	eqtrd 2644	1 ⊢ (𝐴 ∈ On → (𝐹‘𝐴) = (𝐺‘(𝐹 ↾ 𝐴)))

Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977   E cep 4947   ↾ cres 5040  Predcpred 5596  Oncon0 5640  ‘cfv 5804  wrecscwrecs 7293  recscrecs 7354

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-wrecs 7294  df-recs 7355

This theorem is referenced by: (None)