Theorem dfwrecsOLD 8312

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 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (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

Step	Hyp	Ref	Expression
1		df-wrecs 8311	. 2 ⊢ wrecs(𝑅, 𝐴, 𝐹) = frecs(𝑅, 𝐴, (𝐹 ∘ 2nd ))
2		df-frecs 8280	. 2 ⊢ frecs(𝑅, 𝐴, (𝐹 ∘ 2nd )) = ∪ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦(𝐹 ∘ 2nd )(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}
3		vex 3463	. . . . . . . . . . 11 ⊢ 𝑦 ∈ V
4	3	a1i 11	. . . . . . . . . 10 ⊢ (⊤ → 𝑦 ∈ V)
5		vex 3463	. . . . . . . . . . . 12 ⊢ 𝑓 ∈ V
6	5	resex 6016	. . . . . . . . . . 11 ⊢ (𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)) ∈ V
7	6	a1i 11	. . . . . . . . . 10 ⊢ (⊤ → (𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)) ∈ V)
8	4, 7	opco2 8123	. . . . . . . . 9 ⊢ (⊤ → (𝑦(𝐹 ∘ 2nd )(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))
9	8	mptru 1547	. . . . . . . 8 ⊢ (𝑦(𝐹 ∘ 2nd )(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))
10	9	eqeq2i 2748	. . . . . . 7 ⊢ ((𝑓‘𝑦) = (𝑦(𝐹 ∘ 2nd )(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))) ↔ (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))
11	10	ralbii 3082	. . . . . 6 ⊢ (∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦(𝐹 ∘ 2nd )(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))) ↔ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))
12	11	3anbi3i 1159	. . . . 5 ⊢ ((𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦(𝐹 ∘ 2nd )(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))) ↔ (𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))))
13	12	exbii 1848	. . . 4 ⊢ (∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦(𝐹 ∘ 2nd )(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))) ↔ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))))
14	13	abbii 2802	. . 3 ⊢ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦(𝐹 ∘ 2nd )(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))} = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}
15	14	unieqi 4895	. 2 ⊢ ∪ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦(𝐹 ∘ 2nd )(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))} = ∪ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}
16	1, 2, 15	3eqtri 2762	1 ⊢ wrecs(𝑅, 𝐴, 𝐹) = ∪ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}

Syntax hints:   ∧ wa 395   ∧ w3a 1086   = wceq 1540  ⊤wtru 1541  ∃wex 1779   ∈ wcel 2108  {cab 2713  ∀wral 3051  Vcvv 3459   ⊆ wss 3926  ∪ cuni 4883   ↾ cres 5656   ∘ ccom 5658  Predcpred 6289   Fn wfn 6526  ‘cfv 6531  (class class class)co 7405  2^nd c2nd 7987  frecscfrecs 8279  wrecscwrecs 8310

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 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402  ax-un 7729

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-fo 6537  df-fv 6539  df-ov 7408  df-2nd 7989  df-frecs 8280  df-wrecs 8311

This theorem is referenced by:  wrecseq123OLD  8314  nfwrecsOLD  8316  wfrrelOLD  8328  wfrdmssOLD  8329  wfrdmclOLD  8331  wfrfunOLD  8333  wfrlem12OLD  8334  wfrlem16OLD  8338  wfrlem17OLD  8339  dfrecs3OLD  8387