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Theorem dfrecs3 8374
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 8373 . 2 recs(𝐹) = wrecs( E , On, 𝐹)
2 df-wrecs 8299 . 2 wrecs( E , On, 𝐹) = frecs( E , On, (𝐹 ∘ 2nd ))
3 df-frecs 8268 . . 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 3478 . . . . . . . . . . . . 13 𝑥 ∈ V
65elon 6373 . . . . . . . . . . . 12 (𝑥 ∈ On ↔ Ord 𝑥)
7 ordsson 7772 . . . . . . . . . . . . . 14 (Ord 𝑥𝑥 ⊆ On)
8 ordtr 6378 . . . . . . . . . . . . . 14 (Ord 𝑥 → Tr 𝑥)
97, 8jca 512 . . . . . . . . . . . . 13 (Ord 𝑥 → (𝑥 ⊆ On ∧ Tr 𝑥))
10 epweon 7764 . . . . . . . . . . . . . . . 16 E We On
11 wess 5663 . . . . . . . . . . . . . . . 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 6367 . . . . . . . . . . . . . 14 (Ord 𝑥 ↔ (Tr 𝑥 ∧ E We 𝑥))
1513, 14sylibr 233 . . . . . . . . . . . . 13 ((𝑥 ⊆ On ∧ Tr 𝑥) → Ord 𝑥)
169, 15impbii 208 . . . . . . . . . . . 12 (Ord 𝑥 ↔ (𝑥 ⊆ On ∧ Tr 𝑥))
17 dftr3 5271 . . . . . . . . . . . . . 14 (Tr 𝑥 ↔ ∀𝑦𝑥 𝑦𝑥)
18 ssel2 3977 . . . . . . . . . . . . . . . 16 ((𝑥 ⊆ On ∧ 𝑦𝑥) → 𝑦 ∈ On)
19 predon 7775 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ On → Pred( E , On, 𝑦) = 𝑦)
2019sseq1d 4013 . . . . . . . . . . . . . . . 16 (𝑦 ∈ On → (Pred( E , On, 𝑦) ⊆ 𝑥𝑦𝑥))
2118, 20syl 17 . . . . . . . . . . . . . . 15 ((𝑥 ⊆ On ∧ 𝑦𝑥) → (Pred( E , On, 𝑦) ⊆ 𝑥𝑦𝑥))
2221ralbidva 3175 . . . . . . . . . . . . . 14 (𝑥 ⊆ On → (∀𝑦𝑥 Pred( E , On, 𝑦) ⊆ 𝑥 ↔ ∀𝑦𝑥 𝑦𝑥))
2317, 22bitr4id 289 . . . . . . . . . . . . 13 (𝑥 ⊆ On → (Tr 𝑥 ↔ ∀𝑦𝑥 Pred( E , On, 𝑦) ⊆ 𝑥))
2423pm5.32i 575 . . . . . . . . . . . 12 ((𝑥 ⊆ On ∧ Tr 𝑥) ↔ (𝑥 ⊆ On ∧ ∀𝑦𝑥 Pred( E , On, 𝑦) ⊆ 𝑥))
256, 16, 243bitri 296 . . . . . . . . . . 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 6389 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ On ∧ 𝑦𝑥) → 𝑦 ∈ On)
2827, 19syl 17 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ On ∧ 𝑦𝑥) → Pred( E , On, 𝑦) = 𝑦)
2928reseq2d 5981 . . . . . . . . . . . . . . 15 ((𝑥 ∈ On ∧ 𝑦𝑥) → (𝑓 ↾ Pred( E , On, 𝑦)) = (𝑓𝑦))
3029oveq2d 7427 . . . . . . . . . . . . . 14 ((𝑥 ∈ On ∧ 𝑦𝑥) → (𝑦(𝐹 ∘ 2nd )(𝑓 ↾ Pred( E , On, 𝑦))) = (𝑦(𝐹 ∘ 2nd )(𝑓𝑦)))
31 id 22 . . . . . . . . . . . . . . . 16 (𝑦𝑥𝑦𝑥)
32 vex 3478 . . . . . . . . . . . . . . . . . 18 𝑓 ∈ V
3332resex 6029 . . . . . . . . . . . . . . . . 17 (𝑓𝑦) ∈ V
3433a1i 11 . . . . . . . . . . . . . . . 16 (𝑦𝑥 → (𝑓𝑦) ∈ V)
3531, 34opco2 8112 . . . . . . . . . . . . . . 15 (𝑦𝑥 → (𝑦(𝐹 ∘ 2nd )(𝑓𝑦)) = (𝐹‘(𝑓𝑦)))
3635adantl 482 . . . . . . . . . . . . . 14 ((𝑥 ∈ On ∧ 𝑦𝑥) → (𝑦(𝐹 ∘ 2nd )(𝑓𝑦)) = (𝐹‘(𝑓𝑦)))
3730, 36eqtrd 2772 . . . . . . . . . . . . 13 ((𝑥 ∈ On ∧ 𝑦𝑥) → (𝑦(𝐹 ∘ 2nd )(𝑓 ↾ Pred( E , On, 𝑦))) = (𝐹‘(𝑓𝑦)))
3837eqeq2d 2743 . . . . . . . . . . . 12 ((𝑥 ∈ On ∧ 𝑦𝑥) → ((𝑓𝑦) = (𝑦(𝐹 ∘ 2nd )(𝑓 ↾ Pred( E , On, 𝑦))) ↔ (𝑓𝑦) = (𝐹‘(𝑓𝑦))))
3938ralbidva 3175 . . . . . . . . . . 11 (𝑥 ∈ On → (∀𝑦𝑥 (𝑓𝑦) = (𝑦(𝐹 ∘ 2nd )(𝑓 ↾ Pred( E , On, 𝑦))) ↔ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦))))
4039pm5.32i 575 . . . . . . . . . 10 ((𝑥 ∈ On ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦(𝐹 ∘ 2nd )(𝑓 ↾ Pred( E , On, 𝑦)))) ↔ (𝑥 ∈ On ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦))))
4126, 40bitr3i 276 . . . . . . . . 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 643 . . . . . . . 8 ((𝑓 Fn 𝑥 ∧ (𝑥 ∈ On ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))) ↔ (𝑥 ∈ On ∧ (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))))
444, 42, 433bitri 296 . . . . . . 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 1850 . . . . . 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 277 . . . . 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 2802 . . . 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 4921 . . 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 2760 . 2 frecs( E , On, (𝐹 ∘ 2nd )) = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))}
511, 2, 503eqtri 2764 1 recs(𝐹) = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))}
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 396  w3a 1087   = wceq 1541  wex 1781  wcel 2106  {cab 2709  wral 3061  wrex 3070  Vcvv 3474  wss 3948   cuni 4908  Tr wtr 5265   E cep 5579   We wwe 5630  cres 5678  ccom 5680  Predcpred 6299  Ord word 6363  Oncon0 6364   Fn wfn 6538  cfv 6543  (class class class)co 7411  2nd c2nd 7976  frecscfrecs 8267  wrecscwrecs 8298  recscrecs 8372
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fo 6549  df-fv 6551  df-ov 7414  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373
This theorem is referenced by:  recsfval  8383  tfrlem9  8387  dfrdg2  35059  dfrecs2  35214
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