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Theorem tfr3ALT 8442
Description: Alternate proof of tfr3 8439 using well-ordered 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 7806 . . . . . . 7 (𝑥 ∈ On → Pred( E , On, 𝑥) = 𝑥)
21reseq2d 5997 . . . . . 6 (𝑥 ∈ On → (𝐵 ↾ Pred( E , On, 𝑥)) = (𝐵𝑥))
32fveq2d 6910 . . . . 5 (𝑥 ∈ On → (𝐺‘(𝐵 ↾ Pred( E , On, 𝑥))) = (𝐺‘(𝐵𝑥)))
43eqeq2d 2748 . . . 4 (𝑥 ∈ On → ((𝐵𝑥) = (𝐺‘(𝐵 ↾ Pred( E , On, 𝑥))) ↔ (𝐵𝑥) = (𝐺‘(𝐵𝑥))))
54ralbiia 3091 . . 3 (∀𝑥 ∈ On (𝐵𝑥) = (𝐺‘(𝐵 ↾ Pred( E , On, 𝑥))) ↔ ∀𝑥 ∈ On (𝐵𝑥) = (𝐺‘(𝐵𝑥)))
6 epweon 7795 . . . 4 E We On
7 epse 5667 . . . 4 E Se On
8 tfrALT.1 . . . . . 6 𝐹 = recs(𝐺)
9 df-recs 8411 . . . . . 6 recs(𝐺) = wrecs( E , On, 𝐺)
108, 9eqtri 2765 . . . . 5 𝐹 = wrecs( E , On, 𝐺)
1110wfr3 8377 . . . 4 ((( E We On ∧ E Se On) ∧ (𝐵 Fn On ∧ ∀𝑥 ∈ On (𝐵𝑥) = (𝐺‘(𝐵 ↾ Pred( E , On, 𝑥))))) → 𝐹 = 𝐵)
126, 7, 11mpanl12 702 . . 3 ((𝐵 Fn On ∧ ∀𝑥 ∈ On (𝐵𝑥) = (𝐺‘(𝐵 ↾ Pred( E , On, 𝑥)))) → 𝐹 = 𝐵)
135, 12sylan2br 595 . 2 ((𝐵 Fn On ∧ ∀𝑥 ∈ On (𝐵𝑥) = (𝐺‘(𝐵𝑥))) → 𝐹 = 𝐵)
1413eqcomd 2743 1 ((𝐵 Fn On ∧ ∀𝑥 ∈ On (𝐵𝑥) = (𝐺‘(𝐵𝑥))) → 𝐵 = 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  wral 3061   E cep 5583   Se wse 5635   We wwe 5636  cres 5687  Predcpred 6320  Oncon0 6384   Fn wfn 6556  cfv 6561  wrecscwrecs 8336  recscrecs 8410
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411
This theorem is referenced by: (None)
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