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Theorem dfwrecsOLD 8338
Description: Obsolete definition of the well-ordered recursive function generator as of 18-Nov-2024. (New usage is discouraged.) (Proof modification is discouraged.) (Contributed by Scott Fenton, 7-Jun-2018.)
Assertion
Ref Expression
dfwrecsOLD wrecs(𝑅, 𝐴, 𝐹) = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}
Distinct variable groups:   𝑅,𝑓,𝑥,𝑦   𝐴,𝑓,𝑥,𝑦   𝑓,𝐹,𝑥,𝑦

Proof of Theorem dfwrecsOLD
StepHypRef Expression
1 df-wrecs 8337 . 2 wrecs(𝑅, 𝐴, 𝐹) = frecs(𝑅, 𝐴, (𝐹 ∘ 2nd ))
2 df-frecs 8306 . 2 frecs(𝑅, 𝐴, (𝐹 ∘ 2nd )) = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦(𝐹 ∘ 2nd )(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}
3 vex 3484 . . . . . . . . . . 11 𝑦 ∈ V
43a1i 11 . . . . . . . . . 10 (⊤ → 𝑦 ∈ V)
5 vex 3484 . . . . . . . . . . . 12 𝑓 ∈ V
65resex 6047 . . . . . . . . . . 11 (𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)) ∈ V
76a1i 11 . . . . . . . . . 10 (⊤ → (𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)) ∈ V)
84, 7opco2 8149 . . . . . . . . 9 (⊤ → (𝑦(𝐹 ∘ 2nd )(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))
98mptru 1547 . . . . . . . 8 (𝑦(𝐹 ∘ 2nd )(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))
109eqeq2i 2750 . . . . . . 7 ((𝑓𝑦) = (𝑦(𝐹 ∘ 2nd )(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))) ↔ (𝑓𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))
1110ralbii 3093 . . . . . 6 (∀𝑦𝑥 (𝑓𝑦) = (𝑦(𝐹 ∘ 2nd )(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))) ↔ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))
12113anbi3i 1160 . . . . 5 ((𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦(𝐹 ∘ 2nd )(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))) ↔ (𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))))
1312exbii 1848 . . . 4 (∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦(𝐹 ∘ 2nd )(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))) ↔ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))))
1413abbii 2809 . . 3 {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦(𝐹 ∘ 2nd )(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))} = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}
1514unieqi 4919 . 2 {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦(𝐹 ∘ 2nd )(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))} = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}
161, 2, 153eqtri 2769 1 wrecs(𝑅, 𝐴, 𝐹) = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}
Colors of variables: wff setvar class
Syntax hints:  wa 395  w3a 1087   = wceq 1540  wtru 1541  wex 1779  wcel 2108  {cab 2714  wral 3061  Vcvv 3480  wss 3951   cuni 4907  cres 5687  ccom 5689  Predcpred 6320   Fn wfn 6556  cfv 6561  (class class class)co 7431  2nd c2nd 8013  frecscfrecs 8305  wrecscwrecs 8336
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-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-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  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-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
This theorem is referenced by:  wrecseq123OLD  8340  nfwrecsOLD  8342  wfrrelOLD  8354  wfrdmssOLD  8355  wfrdmclOLD  8357  wfrfunOLD  8359  wfrlem12OLD  8360  wfrlem16OLD  8364  wfrlem17OLD  8365  dfrecs3OLD  8413
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