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Theorem dfrecs3 8319
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 8318 . 2 recs(𝐹) = wrecs( E , On, 𝐹)
2 df-wrecs 8244 . 2 wrecs( E , On, 𝐹) = frecs( E , On, (𝐹 ∘ 2nd ))
3 df-frecs 8213 . . 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 1096 . . . . . . . 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 3448 . . . . . . . . . . . . 13 𝑥 ∈ V
65elon 6327 . . . . . . . . . . . 12 (𝑥 ∈ On ↔ Ord 𝑥)
7 ordsson 7718 . . . . . . . . . . . . . 14 (Ord 𝑥𝑥 ⊆ On)
8 ordtr 6332 . . . . . . . . . . . . . 14 (Ord 𝑥 → Tr 𝑥)
97, 8jca 513 . . . . . . . . . . . . 13 (Ord 𝑥 → (𝑥 ⊆ On ∧ Tr 𝑥))
10 epweon 7710 . . . . . . . . . . . . . . . 16 E We On
11 wess 5621 . . . . . . . . . . . . . . . 16 (𝑥 ⊆ On → ( E We On → E We 𝑥))
1210, 11mpi 20 . . . . . . . . . . . . . . 15 (𝑥 ⊆ On → E We 𝑥)
1312anim1ci 617 . . . . . . . . . . . . . 14 ((𝑥 ⊆ On ∧ Tr 𝑥) → (Tr 𝑥 ∧ E We 𝑥))
14 df-ord 6321 . . . . . . . . . . . . . 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 5229 . . . . . . . . . . . . . 14 (Tr 𝑥 ↔ ∀𝑦𝑥 𝑦𝑥)
18 ssel2 3940 . . . . . . . . . . . . . . . 16 ((𝑥 ⊆ On ∧ 𝑦𝑥) → 𝑦 ∈ On)
19 predon 7721 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ On → Pred( E , On, 𝑦) = 𝑦)
2019sseq1d 3976 . . . . . . . . . . . . . . . 16 (𝑦 ∈ On → (Pred( E , On, 𝑦) ⊆ 𝑥𝑦𝑥))
2118, 20syl 17 . . . . . . . . . . . . . . 15 ((𝑥 ⊆ On ∧ 𝑦𝑥) → (Pred( E , On, 𝑦) ⊆ 𝑥𝑦𝑥))
2221ralbidva 3169 . . . . . . . . . . . . . 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 576 . . . . . . . . . . . 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 625 . . . . . . . . . 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 6343 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ On ∧ 𝑦𝑥) → 𝑦 ∈ On)
2827, 19syl 17 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ On ∧ 𝑦𝑥) → Pred( E , On, 𝑦) = 𝑦)
2928reseq2d 5938 . . . . . . . . . . . . . . 15 ((𝑥 ∈ On ∧ 𝑦𝑥) → (𝑓 ↾ Pred( E , On, 𝑦)) = (𝑓𝑦))
3029oveq2d 7374 . . . . . . . . . . . . . 14 ((𝑥 ∈ On ∧ 𝑦𝑥) → (𝑦(𝐹 ∘ 2nd )(𝑓 ↾ Pred( E , On, 𝑦))) = (𝑦(𝐹 ∘ 2nd )(𝑓𝑦)))
31 id 22 . . . . . . . . . . . . . . . 16 (𝑦𝑥𝑦𝑥)
32 vex 3448 . . . . . . . . . . . . . . . . . 18 𝑓 ∈ V
3332resex 5986 . . . . . . . . . . . . . . . . 17 (𝑓𝑦) ∈ V
3433a1i 11 . . . . . . . . . . . . . . . 16 (𝑦𝑥 → (𝑓𝑦) ∈ V)
3531, 34opco2 8057 . . . . . . . . . . . . . . 15 (𝑦𝑥 → (𝑦(𝐹 ∘ 2nd )(𝑓𝑦)) = (𝐹‘(𝑓𝑦)))
3635adantl 483 . . . . . . . . . . . . . 14 ((𝑥 ∈ On ∧ 𝑦𝑥) → (𝑦(𝐹 ∘ 2nd )(𝑓𝑦)) = (𝐹‘(𝑓𝑦)))
3730, 36eqtrd 2773 . . . . . . . . . . . . 13 ((𝑥 ∈ On ∧ 𝑦𝑥) → (𝑦(𝐹 ∘ 2nd )(𝑓 ↾ Pred( E , On, 𝑦))) = (𝐹‘(𝑓𝑦)))
3837eqeq2d 2744 . . . . . . . . . . . 12 ((𝑥 ∈ On ∧ 𝑦𝑥) → ((𝑓𝑦) = (𝑦(𝐹 ∘ 2nd )(𝑓 ↾ Pred( E , On, 𝑦))) ↔ (𝑓𝑦) = (𝐹‘(𝑓𝑦))))
3938ralbidva 3169 . . . . . . . . . . 11 (𝑥 ∈ On → (∀𝑦𝑥 (𝑓𝑦) = (𝑦(𝐹 ∘ 2nd )(𝑓 ↾ Pred( E , On, 𝑦))) ↔ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦))))
4039pm5.32i 576 . . . . . . . . . 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 624 . . . . . . . 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 644 . . . . . . . 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 1851 . . . . . 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 2803 . . . 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 4879 . . 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 2761 . 2 frecs( E , On, (𝐹 ∘ 2nd )) = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))}
511, 2, 503eqtri 2765 1 recs(𝐹) = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))}
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 397  w3a 1088   = wceq 1542  wex 1782  wcel 2107  {cab 2710  wral 3061  wrex 3070  Vcvv 3444  wss 3911   cuni 4866  Tr wtr 5223   E cep 5537   We wwe 5588  cres 5636  ccom 5638  Predcpred 6253  Ord word 6317  Oncon0 6318   Fn wfn 6492  cfv 6497  (class class class)co 7358  2nd c2nd 7921  frecscfrecs 8212  wrecscwrecs 8243  recscrecs 8317
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-fo 6503  df-fv 6505  df-ov 7361  df-2nd 7923  df-frecs 8213  df-wrecs 8244  df-recs 8318
This theorem is referenced by:  recsfval  8328  tfrlem9  8332  dfrdg2  34426  dfrecs2  34581
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