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Theorem tfr3ALT 7385
 Description: Alternate proof of tfr3 7382 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
tfr3ALT ((𝐵 Fn On ∧ ∀𝑥 ∈ On (𝐵𝑥) = (𝐺‘(𝐵𝑥))) → 𝐵 = 𝐹)
Distinct variable groups:   𝑥,𝐵   𝑥,𝐺   𝑥,𝐹

Proof of Theorem tfr3ALT
StepHypRef Expression
1 predon 6883 . . . . . . 7 (𝑥 ∈ On → Pred( E , On, 𝑥) = 𝑥)
21reseq2d 5317 . . . . . 6 (𝑥 ∈ On → (𝐵 ↾ Pred( E , On, 𝑥)) = (𝐵𝑥))
32fveq2d 6107 . . . . 5 (𝑥 ∈ On → (𝐺‘(𝐵 ↾ Pred( E , On, 𝑥))) = (𝐺‘(𝐵𝑥)))
43eqeq2d 2620 . . . 4 (𝑥 ∈ On → ((𝐵𝑥) = (𝐺‘(𝐵 ↾ Pred( E , On, 𝑥))) ↔ (𝐵𝑥) = (𝐺‘(𝐵𝑥))))
54ralbiia 2962 . . 3 (∀𝑥 ∈ On (𝐵𝑥) = (𝐺‘(𝐵 ↾ Pred( E , On, 𝑥))) ↔ ∀𝑥 ∈ On (𝐵𝑥) = (𝐺‘(𝐵𝑥)))
6 epweon 6875 . . . 4 E We On
7 epse 5021 . . . 4 E Se On
8 tfrALT.1 . . . . 5 𝐹 = recs(𝐺)
9 df-recs 7355 . . . . 5 recs(𝐺) = wrecs( E , On, 𝐺)
108, 9eqtri 2632 . . . 4 𝐹 = wrecs( E , On, 𝐺)
116, 7, 10wfr3 7322 . . 3 ((𝐵 Fn On ∧ ∀𝑥 ∈ On (𝐵𝑥) = (𝐺‘(𝐵 ↾ Pred( E , On, 𝑥)))) → 𝐹 = 𝐵)
125, 11sylan2br 492 . 2 ((𝐵 Fn On ∧ ∀𝑥 ∈ On (𝐵𝑥) = (𝐺‘(𝐵𝑥))) → 𝐹 = 𝐵)
1312eqcomd 2616 1 ((𝐵 Fn On ∧ ∀𝑥 ∈ On (𝐵𝑥) = (𝐺‘(𝐵𝑥))) → 𝐵 = 𝐹)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896   E cep 4947   ↾ cres 5040  Predcpred 5596  Oncon0 5640   Fn wfn 5799  ‘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)
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