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Theorem dfrecs3 8412
Description: The old definition of transfinite recursion. This version is preferred for development, as it demonstrates the properties of transfinite recursion without relying on well-ordered recursion. (Contributed by Scott Fenton, 3-Aug-2020.) (Proof revised by Scott Fenton, 18-Nov-2024.)
Assertion
Ref Expression
dfrecs3 recs(𝐹) = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))}
Distinct variable group:   𝑓,𝐹,𝑥,𝑦

Proof of Theorem dfrecs3
StepHypRef Expression
1 df-recs 8411 . 2 recs(𝐹) = wrecs( E , On, 𝐹)
2 df-wrecs 8337 . 2 wrecs( E , On, 𝐹) = frecs( E , On, (𝐹 ∘ 2nd ))
3 df-frecs 8306 . . 3 frecs( E , On, (𝐹 ∘ 2nd )) = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ On ∧ ∀𝑦𝑥 Pred( E , On, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦(𝐹 ∘ 2nd )(𝑓 ↾ Pred( E , On, 𝑦))))}
4 3anass 1095 . . . . . . . 8 ((𝑓 Fn 𝑥 ∧ (𝑥 ⊆ On ∧ ∀𝑦𝑥 Pred( E , On, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦(𝐹 ∘ 2nd )(𝑓 ↾ Pred( E , On, 𝑦)))) ↔ (𝑓 Fn 𝑥 ∧ ((𝑥 ⊆ On ∧ ∀𝑦𝑥 Pred( E , On, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦(𝐹 ∘ 2nd )(𝑓 ↾ Pred( E , On, 𝑦))))))
5 vex 3484 . . . . . . . . . . . . 13 𝑥 ∈ V
65elon 6393 . . . . . . . . . . . 12 (𝑥 ∈ On ↔ Ord 𝑥)
7 ordsson 7803 . . . . . . . . . . . . . 14 (Ord 𝑥𝑥 ⊆ On)
8 ordtr 6398 . . . . . . . . . . . . . 14 (Ord 𝑥 → Tr 𝑥)
97, 8jca 511 . . . . . . . . . . . . 13 (Ord 𝑥 → (𝑥 ⊆ On ∧ Tr 𝑥))
10 epweon 7795 . . . . . . . . . . . . . . . 16 E We On
11 wess 5671 . . . . . . . . . . . . . . . 16 (𝑥 ⊆ On → ( E We On → E We 𝑥))
1210, 11mpi 20 . . . . . . . . . . . . . . 15 (𝑥 ⊆ On → E We 𝑥)
1312anim1ci 616 . . . . . . . . . . . . . 14 ((𝑥 ⊆ On ∧ Tr 𝑥) → (Tr 𝑥 ∧ E We 𝑥))
14 df-ord 6387 . . . . . . . . . . . . . 14 (Ord 𝑥 ↔ (Tr 𝑥 ∧ E We 𝑥))
1513, 14sylibr 234 . . . . . . . . . . . . 13 ((𝑥 ⊆ On ∧ Tr 𝑥) → Ord 𝑥)
169, 15impbii 209 . . . . . . . . . . . 12 (Ord 𝑥 ↔ (𝑥 ⊆ On ∧ Tr 𝑥))
17 dftr3 5265 . . . . . . . . . . . . . 14 (Tr 𝑥 ↔ ∀𝑦𝑥 𝑦𝑥)
18 ssel2 3978 . . . . . . . . . . . . . . . 16 ((𝑥 ⊆ On ∧ 𝑦𝑥) → 𝑦 ∈ On)
19 predon 7806 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ On → Pred( E , On, 𝑦) = 𝑦)
2019sseq1d 4015 . . . . . . . . . . . . . . . 16 (𝑦 ∈ On → (Pred( E , On, 𝑦) ⊆ 𝑥𝑦𝑥))
2118, 20syl 17 . . . . . . . . . . . . . . 15 ((𝑥 ⊆ On ∧ 𝑦𝑥) → (Pred( E , On, 𝑦) ⊆ 𝑥𝑦𝑥))
2221ralbidva 3176 . . . . . . . . . . . . . 14 (𝑥 ⊆ On → (∀𝑦𝑥 Pred( E , On, 𝑦) ⊆ 𝑥 ↔ ∀𝑦𝑥 𝑦𝑥))
2317, 22bitr4id 290 . . . . . . . . . . . . 13 (𝑥 ⊆ On → (Tr 𝑥 ↔ ∀𝑦𝑥 Pred( E , On, 𝑦) ⊆ 𝑥))
2423pm5.32i 574 . . . . . . . . . . . 12 ((𝑥 ⊆ On ∧ Tr 𝑥) ↔ (𝑥 ⊆ On ∧ ∀𝑦𝑥 Pred( E , On, 𝑦) ⊆ 𝑥))
256, 16, 243bitri 297 . . . . . . . . . . 11 (𝑥 ∈ On ↔ (𝑥 ⊆ On ∧ ∀𝑦𝑥 Pred( E , On, 𝑦) ⊆ 𝑥))
2625anbi1i 624 . . . . . . . . . 10 ((𝑥 ∈ On ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦(𝐹 ∘ 2nd )(𝑓 ↾ Pred( E , On, 𝑦)))) ↔ ((𝑥 ⊆ On ∧ ∀𝑦𝑥 Pred( E , On, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦(𝐹 ∘ 2nd )(𝑓 ↾ Pred( E , On, 𝑦)))))
27 onelon 6409 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ On ∧ 𝑦𝑥) → 𝑦 ∈ On)
2827, 19syl 17 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ On ∧ 𝑦𝑥) → Pred( E , On, 𝑦) = 𝑦)
2928reseq2d 5997 . . . . . . . . . . . . . . 15 ((𝑥 ∈ On ∧ 𝑦𝑥) → (𝑓 ↾ Pred( E , On, 𝑦)) = (𝑓𝑦))
3029oveq2d 7447 . . . . . . . . . . . . . 14 ((𝑥 ∈ On ∧ 𝑦𝑥) → (𝑦(𝐹 ∘ 2nd )(𝑓 ↾ Pred( E , On, 𝑦))) = (𝑦(𝐹 ∘ 2nd )(𝑓𝑦)))
31 id 22 . . . . . . . . . . . . . . . 16 (𝑦𝑥𝑦𝑥)
32 vex 3484 . . . . . . . . . . . . . . . . . 18 𝑓 ∈ V
3332resex 6047 . . . . . . . . . . . . . . . . 17 (𝑓𝑦) ∈ V
3433a1i 11 . . . . . . . . . . . . . . . 16 (𝑦𝑥 → (𝑓𝑦) ∈ V)
3531, 34opco2 8149 . . . . . . . . . . . . . . 15 (𝑦𝑥 → (𝑦(𝐹 ∘ 2nd )(𝑓𝑦)) = (𝐹‘(𝑓𝑦)))
3635adantl 481 . . . . . . . . . . . . . 14 ((𝑥 ∈ On ∧ 𝑦𝑥) → (𝑦(𝐹 ∘ 2nd )(𝑓𝑦)) = (𝐹‘(𝑓𝑦)))
3730, 36eqtrd 2777 . . . . . . . . . . . . 13 ((𝑥 ∈ On ∧ 𝑦𝑥) → (𝑦(𝐹 ∘ 2nd )(𝑓 ↾ Pred( E , On, 𝑦))) = (𝐹‘(𝑓𝑦)))
3837eqeq2d 2748 . . . . . . . . . . . 12 ((𝑥 ∈ On ∧ 𝑦𝑥) → ((𝑓𝑦) = (𝑦(𝐹 ∘ 2nd )(𝑓 ↾ Pred( E , On, 𝑦))) ↔ (𝑓𝑦) = (𝐹‘(𝑓𝑦))))
3938ralbidva 3176 . . . . . . . . . . 11 (𝑥 ∈ On → (∀𝑦𝑥 (𝑓𝑦) = (𝑦(𝐹 ∘ 2nd )(𝑓 ↾ Pred( E , On, 𝑦))) ↔ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦))))
4039pm5.32i 574 . . . . . . . . . 10 ((𝑥 ∈ On ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦(𝐹 ∘ 2nd )(𝑓 ↾ Pred( E , On, 𝑦)))) ↔ (𝑥 ∈ On ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦))))
4126, 40bitr3i 277 . . . . . . . . 9 (((𝑥 ⊆ On ∧ ∀𝑦𝑥 Pred( E , On, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦(𝐹 ∘ 2nd )(𝑓 ↾ Pred( E , On, 𝑦)))) ↔ (𝑥 ∈ On ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦))))
4241anbi2i 623 . . . . . . . 8 ((𝑓 Fn 𝑥 ∧ ((𝑥 ⊆ On ∧ ∀𝑦𝑥 Pred( E , On, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦(𝐹 ∘ 2nd )(𝑓 ↾ Pred( E , On, 𝑦))))) ↔ (𝑓 Fn 𝑥 ∧ (𝑥 ∈ On ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))))
43 an12 645 . . . . . . . 8 ((𝑓 Fn 𝑥 ∧ (𝑥 ∈ On ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))) ↔ (𝑥 ∈ On ∧ (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))))
444, 42, 433bitri 297 . . . . . . 7 ((𝑓 Fn 𝑥 ∧ (𝑥 ⊆ On ∧ ∀𝑦𝑥 Pred( E , On, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦(𝐹 ∘ 2nd )(𝑓 ↾ Pred( E , On, 𝑦)))) ↔ (𝑥 ∈ On ∧ (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))))
4544exbii 1848 . . . . . 6 (∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ On ∧ ∀𝑦𝑥 Pred( E , On, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦(𝐹 ∘ 2nd )(𝑓 ↾ Pred( E , On, 𝑦)))) ↔ ∃𝑥(𝑥 ∈ On ∧ (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))))
46 df-rex 3071 . . . . . 6 (∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦))) ↔ ∃𝑥(𝑥 ∈ On ∧ (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))))
4745, 46bitr4i 278 . . . . 5 (∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ On ∧ ∀𝑦𝑥 Pred( E , On, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦(𝐹 ∘ 2nd )(𝑓 ↾ Pred( E , On, 𝑦)))) ↔ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦))))
4847abbii 2809 . . . 4 {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ On ∧ ∀𝑦𝑥 Pred( E , On, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦(𝐹 ∘ 2nd )(𝑓 ↾ Pred( E , On, 𝑦))))} = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))}
4948unieqi 4919 . . 3 {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ On ∧ ∀𝑦𝑥 Pred( E , On, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦(𝐹 ∘ 2nd )(𝑓 ↾ Pred( E , On, 𝑦))))} = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))}
503, 49eqtri 2765 . 2 frecs( E , On, (𝐹 ∘ 2nd )) = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))}
511, 2, 503eqtri 2769 1 recs(𝐹) = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))}
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395  w3a 1087   = wceq 1540  wex 1779  wcel 2108  {cab 2714  wral 3061  wrex 3070  Vcvv 3480  wss 3951   cuni 4907  Tr wtr 5259   E cep 5583   We wwe 5636  cres 5687  ccom 5689  Predcpred 6320  Ord word 6383  Oncon0 6384   Fn wfn 6556  cfv 6561  (class class class)co 7431  2nd c2nd 8013  frecscfrecs 8305  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-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-rab 3437  df-v 3482  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-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-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-fo 6567  df-fv 6569  df-ov 7434  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411
This theorem is referenced by:  recsfval  8421  tfrlem9  8425  dfrdg2  35796  dfrecs2  35951
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