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Theorem tfr3ALT 7702
Description: Alternate proof of tfr3 7699 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 7189 . . . . . . 7 (𝑥 ∈ On → Pred( E , On, 𝑥) = 𝑥)
21reseq2d 5565 . . . . . 6 (𝑥 ∈ On → (𝐵 ↾ Pred( E , On, 𝑥)) = (𝐵𝑥))
32fveq2d 6379 . . . . 5 (𝑥 ∈ On → (𝐺‘(𝐵 ↾ Pred( E , On, 𝑥))) = (𝐺‘(𝐵𝑥)))
43eqeq2d 2775 . . . 4 (𝑥 ∈ On → ((𝐵𝑥) = (𝐺‘(𝐵 ↾ Pred( E , On, 𝑥))) ↔ (𝐵𝑥) = (𝐺‘(𝐵𝑥))))
54ralbiia 3126 . . 3 (∀𝑥 ∈ On (𝐵𝑥) = (𝐺‘(𝐵 ↾ Pred( E , On, 𝑥))) ↔ ∀𝑥 ∈ On (𝐵𝑥) = (𝐺‘(𝐵𝑥)))
6 epweon 7181 . . . 4 E We On
7 epse 5260 . . . 4 E Se On
8 tfrALT.1 . . . . 5 𝐹 = recs(𝐺)
9 df-recs 7672 . . . . 5 recs(𝐺) = wrecs( E , On, 𝐺)
108, 9eqtri 2787 . . . 4 𝐹 = wrecs( E , On, 𝐺)
116, 7, 10wfr3 7639 . . 3 ((𝐵 Fn On ∧ ∀𝑥 ∈ On (𝐵𝑥) = (𝐺‘(𝐵 ↾ Pred( E , On, 𝑥)))) → 𝐹 = 𝐵)
125, 11sylan2br 588 . 2 ((𝐵 Fn On ∧ ∀𝑥 ∈ On (𝐵𝑥) = (𝐺‘(𝐵𝑥))) → 𝐹 = 𝐵)
1312eqcomd 2771 1 ((𝐵 Fn On ∧ ∀𝑥 ∈ On (𝐵𝑥) = (𝐺‘(𝐵𝑥))) → 𝐵 = 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1652  wcel 2155  wral 3055   E cep 5189  cres 5279  Predcpred 5864  Oncon0 5908   Fn wfn 6063  cfv 6068  wrecscwrecs 7609  recscrecs 7671
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-se 5237  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-wrecs 7610  df-recs 7672
This theorem is referenced by: (None)
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