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Theorem dfwrecsOLD 8245
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 8244 . 2 wrecs(𝑅, 𝐴, 𝐹) = frecs(𝑅, 𝐴, (𝐹 ∘ 2nd ))
2 df-frecs 8213 . 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 3448 . . . . . . . . . . 11 𝑦 ∈ V
43a1i 11 . . . . . . . . . 10 (⊤ → 𝑦 ∈ V)
5 vex 3448 . . . . . . . . . . . 12 𝑓 ∈ V
65resex 5986 . . . . . . . . . . 11 (𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)) ∈ V
76a1i 11 . . . . . . . . . 10 (⊤ → (𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)) ∈ V)
84, 7opco2 8057 . . . . . . . . 9 (⊤ → (𝑦(𝐹 ∘ 2nd )(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))
98mptru 1549 . . . . . . . 8 (𝑦(𝐹 ∘ 2nd )(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))
109eqeq2i 2746 . . . . . . 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 1851 . . . 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 2803 . . 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 4879 . 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 2765 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 397  w3a 1088   = wceq 1542  wtru 1543  wex 1782  wcel 2107  {cab 2710  wral 3061  Vcvv 3444  wss 3911   cuni 4866  cres 5636  ccom 5638  Predcpred 6253   Fn wfn 6492  cfv 6497  (class class class)co 7358  2nd c2nd 7921  frecscfrecs 8212  wrecscwrecs 8243
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-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-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  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-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
This theorem is referenced by:  wrecseq123OLD  8247  nfwrecsOLD  8249  wfrrelOLD  8261  wfrdmssOLD  8262  wfrdmclOLD  8264  wfrfunOLD  8266  wfrlem12OLD  8267  wfrlem16OLD  8271  wfrlem17OLD  8272  dfrecs3OLD  8320
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