Theorem dfrecs3 8359

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 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))}

Distinct variable group:   𝑓,𝐹,𝑥,𝑦

Proof of Theorem dfrecs3

Step	Hyp	Ref	Expression
1		df-recs 8358	. 2 ⊢ recs(𝐹) = wrecs( E , On, 𝐹)
2		df-wrecs 8309	. 2 ⊢ wrecs( E , On, 𝐹) = frecs( E , On, (𝐹 ∘ 2nd ))
3		df-frecs 8278	. . 3 ⊢ frecs( E , On, (𝐹 ∘ 2nd )) = ∪ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ On ∧ ∀𝑦 ∈ 𝑥 Pred( E , On, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦(𝐹 ∘ 2nd )(𝑓 ↾ Pred( E , On, 𝑦))))}
4		3anass 1109	. . . . . . . 8 ⊢ ((𝑓 Fn 𝑥 ∧ (𝑥 ⊆ On ∧ ∀𝑦 ∈ 𝑥 Pred( E , On, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦(𝐹 ∘ 2nd )(𝑓 ↾ Pred( E , On, 𝑦)))) ↔ (𝑓 Fn 𝑥 ∧ ((𝑥 ⊆ On ∧ ∀𝑦 ∈ 𝑥 Pred( E , On, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦(𝐹 ∘ 2nd )(𝑓 ↾ Pred( E , On, 𝑦))))))
5		vex 3465	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ V
6	5	elon 6370	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ On ↔ Ord 𝑥)
7		ordsson 7782	. . . . . . . . . . . . . 14 ⊢ (Ord 𝑥 → 𝑥 ⊆ On)
8		ordtr 6375	. . . . . . . . . . . . . 14 ⊢ (Ord 𝑥 → Tr 𝑥)
9	7, 8	jca 520	. . . . . . . . . . . . 13 ⊢ (Ord 𝑥 → (𝑥 ⊆ On ∧ Tr 𝑥))
10		epweon 7774	. . . . . . . . . . . . . . . 16 ⊢ E We On
11		wess 5648	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ⊆ On → ( E We On → E We 𝑥))
12	10, 11	mpi 21	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ⊆ On → E We 𝑥)
13	12	anim1ci 627	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ⊆ On ∧ Tr 𝑥) → (Tr 𝑥 ∧ E We 𝑥))
14		df-ord 6364	. . . . . . . . . . . . . 14 ⊢ (Ord 𝑥 ↔ (Tr 𝑥 ∧ E We 𝑥))
15	13, 14	sylibr 237	. . . . . . . . . . . . 13 ⊢ ((𝑥 ⊆ On ∧ Tr 𝑥) → Ord 𝑥)
16	9, 15	impbii 212	. . . . . . . . . . . 12 ⊢ (Ord 𝑥 ↔ (𝑥 ⊆ On ∧ Tr 𝑥))
17		dftr3 5225	. . . . . . . . . . . . . 14 ⊢ (Tr 𝑥 ↔ ∀𝑦 ∈ 𝑥 𝑦 ⊆ 𝑥)
18		ssel2 3938	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ⊆ On ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ On)
19		predon 7785	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ On → Pred( E , On, 𝑦) = 𝑦)
20	19	sseq1d 3974	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ On → (Pred( E , On, 𝑦) ⊆ 𝑥 ↔ 𝑦 ⊆ 𝑥))
21	18, 20	syl 18	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ⊆ On ∧ 𝑦 ∈ 𝑥) → (Pred( E , On, 𝑦) ⊆ 𝑥 ↔ 𝑦 ⊆ 𝑥))
22	21	ralbidva 3192	. . . . . . . . . . . . . 14 ⊢ (𝑥 ⊆ On → (∀𝑦 ∈ 𝑥 Pred( E , On, 𝑦) ⊆ 𝑥 ↔ ∀𝑦 ∈ 𝑥 𝑦 ⊆ 𝑥))
23	17, 22	bitr4id 293	. . . . . . . . . . . . 13 ⊢ (𝑥 ⊆ On → (Tr 𝑥 ↔ ∀𝑦 ∈ 𝑥 Pred( E , On, 𝑦) ⊆ 𝑥))
24	23	pm5.32i 584	. . . . . . . . . . . 12 ⊢ ((𝑥 ⊆ On ∧ Tr 𝑥) ↔ (𝑥 ⊆ On ∧ ∀𝑦 ∈ 𝑥 Pred( E , On, 𝑦) ⊆ 𝑥))
25	6, 16, 24	3bitri 300	. . . . . . . . . . 11 ⊢ (𝑥 ∈ On ↔ (𝑥 ⊆ On ∧ ∀𝑦 ∈ 𝑥 Pred( E , On, 𝑦) ⊆ 𝑥))
26	25	anbi1i 635	. . . . . . . . . 10 ⊢ ((𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦(𝐹 ∘ 2nd )(𝑓 ↾ Pred( E , On, 𝑦)))) ↔ ((𝑥 ⊆ On ∧ ∀𝑦 ∈ 𝑥 Pred( E , On, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦(𝐹 ∘ 2nd )(𝑓 ↾ Pred( E , On, 𝑦)))))
27		onelon 6386	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ On)
28	27, 19	syl 18	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → Pred( E , On, 𝑦) = 𝑦)
29	28	reseq2d 5979	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → (𝑓 ↾ Pred( E , On, 𝑦)) = (𝑓 ↾ 𝑦))
30	29	oveq2d 7427	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → (𝑦(𝐹 ∘ 2nd )(𝑓 ↾ Pred( E , On, 𝑦))) = (𝑦(𝐹 ∘ 2nd )(𝑓 ↾ 𝑦)))
31		id 23	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ 𝑥 → 𝑦 ∈ 𝑥)
32		vex 3465	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑓 ∈ V
33	32	resex 6029	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 ↾ 𝑦) ∈ V
34	33	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ 𝑥 → (𝑓 ↾ 𝑦) ∈ V)
35	31, 34	opco2 8119	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ 𝑥 → (𝑦(𝐹 ∘ 2nd )(𝑓 ↾ 𝑦)) = (𝐹‘(𝑓 ↾ 𝑦)))
36	35	adantl 486	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → (𝑦(𝐹 ∘ 2nd )(𝑓 ↾ 𝑦)) = (𝐹‘(𝑓 ↾ 𝑦)))
37	30, 36	eqtrd 2804	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → (𝑦(𝐹 ∘ 2nd )(𝑓 ↾ Pred( E , On, 𝑦))) = (𝐹‘(𝑓 ↾ 𝑦)))
38	37	eqeq2d 2780	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → ((𝑓‘𝑦) = (𝑦(𝐹 ∘ 2nd )(𝑓 ↾ Pred( E , On, 𝑦))) ↔ (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦))))
39	38	ralbidva 3192	. . . . . . . . . . 11 ⊢ (𝑥 ∈ On → (∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦(𝐹 ∘ 2nd )(𝑓 ↾ Pred( E , On, 𝑦))) ↔ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦))))
40	39	pm5.32i 584	. . . . . . . . . 10 ⊢ ((𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦(𝐹 ∘ 2nd )(𝑓 ↾ Pred( E , On, 𝑦)))) ↔ (𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦))))
41	26, 40	bitr3i 280	. . . . . . . . 9 ⊢ (((𝑥 ⊆ On ∧ ∀𝑦 ∈ 𝑥 Pred( E , On, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦(𝐹 ∘ 2nd )(𝑓 ↾ Pred( E , On, 𝑦)))) ↔ (𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦))))
42	41	anbi2i 634	. . . . . . . 8 ⊢ ((𝑓 Fn 𝑥 ∧ ((𝑥 ⊆ On ∧ ∀𝑦 ∈ 𝑥 Pred( E , On, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦(𝐹 ∘ 2nd )(𝑓 ↾ Pred( E , On, 𝑦))))) ↔ (𝑓 Fn 𝑥 ∧ (𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))))
43		an12 657	. . . . . . . 8 ⊢ ((𝑓 Fn 𝑥 ∧ (𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))) ↔ (𝑥 ∈ On ∧ (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))))
44	4, 42, 43	3bitri 300	. . . . . . 7 ⊢ ((𝑓 Fn 𝑥 ∧ (𝑥 ⊆ On ∧ ∀𝑦 ∈ 𝑥 Pred( E , On, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦(𝐹 ∘ 2nd )(𝑓 ↾ Pred( E , On, 𝑦)))) ↔ (𝑥 ∈ On ∧ (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))))
45	44	exbii 1875	. . . . . 6 ⊢ (∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ On ∧ ∀𝑦 ∈ 𝑥 Pred( E , On, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦(𝐹 ∘ 2nd )(𝑓 ↾ Pred( E , On, 𝑦)))) ↔ ∃𝑥(𝑥 ∈ On ∧ (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))))
46		df-rex 3096	. . . . . 6 ⊢ (∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦))) ↔ ∃𝑥(𝑥 ∈ On ∧ (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))))
47	45, 46	bitr4i 281	. . . . 5 ⊢ (∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ On ∧ ∀𝑦 ∈ 𝑥 Pred( E , On, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦(𝐹 ∘ 2nd )(𝑓 ↾ Pred( E , On, 𝑦)))) ↔ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦))))
48	47	abbii 2836	. . . 4 ⊢ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ On ∧ ∀𝑦 ∈ 𝑥 Pred( E , On, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦(𝐹 ∘ 2nd )(𝑓 ↾ Pred( E , On, 𝑦))))} = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))}
49	48	unieqi 4886	. . 3 ⊢ ∪ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ On ∧ ∀𝑦 ∈ 𝑥 Pred( E , On, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦(𝐹 ∘ 2nd )(𝑓 ↾ Pred( E , On, 𝑦))))} = ∪ {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))}
50	3, 49	eqtri 2792	. 2 ⊢ frecs( E , On, (𝐹 ∘ 2nd )) = ∪ {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))}
51	1, 2, 50	3eqtri 2796	1 ⊢ recs(𝐹) = ∪ {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))}

Syntax hints:   ↔ wb 209   ∧ wa 400   ∧ w3a 1101   = wceq 1567  ∃wex 1806   ∈ wcel 2149  {cab 2747  ∀wral 3085  ∃wrex 3095  Vcvv 3461   ⊆ wss 3911  ∪ cuni 4874  Tr wtr 5220   E cep 5561   We wwe 5614   ↾ cres 5664   ∘ ccom 5666  Predcpred 6302  Ord word 6360  Oncon0 6361   Fn wfn 6532  ‘cfv 6537  (class class class)co 7411  2^nd c2nd 7985  frecscfrecs 8277  wrecscwrecs 8308  recscrecs 8357

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5259  ax-nul 5271  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3423  df-v 3463  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4491  df-pw 4567  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-pred 6303  df-ord 6364  df-on 6365  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-fo 6543  df-fv 6545  df-ov 7414  df-2nd 7987  df-frecs 8278  df-wrecs 8309  df-recs 8358

This theorem is referenced by:  recsfval  8367  tfrlem9  8372  dfrdg2  36218  dfrecs2  36375