Theorem dfwrecsOLD 8327

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 8326	. 2 ⊢ wrecs(𝑅, 𝐴, 𝐹) = frecs(𝑅, 𝐴, (𝐹 ∘ 2nd ))
2		df-frecs 8295	. 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 3465	. . . . . . . . . . 11 ⊢ 𝑦 ∈ V
4	3	a1i 11	. . . . . . . . . 10 ⊢ (⊤ → 𝑦 ∈ V)
5		vex 3465	. . . . . . . . . . . 12 ⊢ 𝑓 ∈ V
6	5	resex 6037	. . . . . . . . . . 11 ⊢ (𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)) ∈ V
7	6	a1i 11	. . . . . . . . . 10 ⊢ (⊤ → (𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)) ∈ V)
8	4, 7	opco2 8137	. . . . . . . . 9 ⊢ (⊤ → (𝑦(𝐹 ∘ 2nd )(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))
9	8	mptru 1540	. . . . . . . 8 ⊢ (𝑦(𝐹 ∘ 2nd )(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))
10	9	eqeq2i 2738	. . . . . . 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 1156	. . . . 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 1842	. . . 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 2795	. . 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 4924	. 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 2757	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 394   ∧ w3a 1084   = wceq 1533  ⊤wtru 1534  ∃wex 1773   ∈ wcel 2098  {cab 2702  ∀wral 3050  Vcvv 3461   ⊆ wss 3946  ∪ cuni 4912   ↾ cres 5683   ∘ ccom 5685  Predcpred 6310   Fn wfn 6548  ‘cfv 6553  (class class class)co 7423  2^nd c2nd 8001  frecscfrecs 8294  wrecscwrecs 8325

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5303  ax-nul 5310  ax-pr 5432  ax-un 7745

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4325  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5579  df-xp 5687  df-rel 5688  df-cnv 5689  df-co 5690  df-dm 5691  df-rn 5692  df-res 5693  df-ima 5694  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-fo 6559  df-fv 6561  df-ov 7426  df-2nd 8003  df-frecs 8295  df-wrecs 8326

This theorem is referenced by:  wrecseq123OLD  8329  nfwrecsOLD  8331  wfrrelOLD  8343  wfrdmssOLD  8344  wfrdmclOLD  8346  wfrfunOLD  8348  wfrlem12OLD  8349  wfrlem16OLD  8353  wfrlem17OLD  8354  dfrecs3OLD  8402