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Theorem tfr3ALT 7668
Description: Alternate proof of tfr3 7665 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 7157 . . . . . . 7 (𝑥 ∈ On → Pred( E , On, 𝑥) = 𝑥)
21reseq2d 5551 . . . . . 6 (𝑥 ∈ On → (𝐵 ↾ Pred( E , On, 𝑥)) = (𝐵𝑥))
32fveq2d 6357 . . . . 5 (𝑥 ∈ On → (𝐺‘(𝐵 ↾ Pred( E , On, 𝑥))) = (𝐺‘(𝐵𝑥)))
43eqeq2d 2770 . . . 4 (𝑥 ∈ On → ((𝐵𝑥) = (𝐺‘(𝐵 ↾ Pred( E , On, 𝑥))) ↔ (𝐵𝑥) = (𝐺‘(𝐵𝑥))))
54ralbiia 3117 . . 3 (∀𝑥 ∈ On (𝐵𝑥) = (𝐺‘(𝐵 ↾ Pred( E , On, 𝑥))) ↔ ∀𝑥 ∈ On (𝐵𝑥) = (𝐺‘(𝐵𝑥)))
6 epweon 7149 . . . 4 E We On
7 epse 5249 . . . 4 E Se On
8 tfrALT.1 . . . . 5 𝐹 = recs(𝐺)
9 df-recs 7638 . . . . 5 recs(𝐺) = wrecs( E , On, 𝐺)
108, 9eqtri 2782 . . . 4 𝐹 = wrecs( E , On, 𝐺)
116, 7, 10wfr3 7605 . . 3 ((𝐵 Fn On ∧ ∀𝑥 ∈ On (𝐵𝑥) = (𝐺‘(𝐵 ↾ Pred( E , On, 𝑥)))) → 𝐹 = 𝐵)
125, 11sylan2br 494 . 2 ((𝐵 Fn On ∧ ∀𝑥 ∈ On (𝐵𝑥) = (𝐺‘(𝐵𝑥))) → 𝐹 = 𝐵)
1312eqcomd 2766 1 ((𝐵 Fn On ∧ ∀𝑥 ∈ On (𝐵𝑥) = (𝐺‘(𝐵𝑥))) → 𝐵 = 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  wcel 2139  wral 3050   E cep 5178  cres 5268  Predcpred 5840  Oncon0 5884   Fn wfn 6044  cfv 6049  wrecscwrecs 7576  recscrecs 7637
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-se 5226  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-wrecs 7577  df-recs 7638
This theorem is referenced by: (None)
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