Theorem dfrecs3 8374

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 8373	. 2 ⊢ recs(𝐹) = wrecs( E , On, 𝐹)
2		df-wrecs 8299	. 2 ⊢ wrecs( E , On, 𝐹) = frecs( E , On, (𝐹 ∘ 2nd ))
3		df-frecs 8268	. . 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 1093	. . . . . . . 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 3476	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ V
6	5	elon 6372	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ On ↔ Ord 𝑥)
7		ordsson 7772	. . . . . . . . . . . . . 14 ⊢ (Ord 𝑥 → 𝑥 ⊆ On)
8		ordtr 6377	. . . . . . . . . . . . . 14 ⊢ (Ord 𝑥 → Tr 𝑥)
9	7, 8	jca 510	. . . . . . . . . . . . 13 ⊢ (Ord 𝑥 → (𝑥 ⊆ On ∧ Tr 𝑥))
10		epweon 7764	. . . . . . . . . . . . . . . 16 ⊢ E We On
11		wess 5662	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ⊆ On → ( E We On → E We 𝑥))
12	10, 11	mpi 20	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ⊆ On → E We 𝑥)
13	12	anim1ci 614	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ⊆ On ∧ Tr 𝑥) → (Tr 𝑥 ∧ E We 𝑥))
14		df-ord 6366	. . . . . . . . . . . . . 14 ⊢ (Ord 𝑥 ↔ (Tr 𝑥 ∧ E We 𝑥))
15	13, 14	sylibr 233	. . . . . . . . . . . . 13 ⊢ ((𝑥 ⊆ On ∧ Tr 𝑥) → Ord 𝑥)
16	9, 15	impbii 208	. . . . . . . . . . . 12 ⊢ (Ord 𝑥 ↔ (𝑥 ⊆ On ∧ Tr 𝑥))
17		dftr3 5270	. . . . . . . . . . . . . 14 ⊢ (Tr 𝑥 ↔ ∀𝑦 ∈ 𝑥 𝑦 ⊆ 𝑥)
18		ssel2 3976	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ⊆ On ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ On)
19		predon 7775	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ On → Pred( E , On, 𝑦) = 𝑦)
20	19	sseq1d 4012	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ On → (Pred( E , On, 𝑦) ⊆ 𝑥 ↔ 𝑦 ⊆ 𝑥))
21	18, 20	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ⊆ On ∧ 𝑦 ∈ 𝑥) → (Pred( E , On, 𝑦) ⊆ 𝑥 ↔ 𝑦 ⊆ 𝑥))
22	21	ralbidva 3173	. . . . . . . . . . . . . 14 ⊢ (𝑥 ⊆ On → (∀𝑦 ∈ 𝑥 Pred( E , On, 𝑦) ⊆ 𝑥 ↔ ∀𝑦 ∈ 𝑥 𝑦 ⊆ 𝑥))
23	17, 22	bitr4id 289	. . . . . . . . . . . . 13 ⊢ (𝑥 ⊆ On → (Tr 𝑥 ↔ ∀𝑦 ∈ 𝑥 Pred( E , On, 𝑦) ⊆ 𝑥))
24	23	pm5.32i 573	. . . . . . . . . . . 12 ⊢ ((𝑥 ⊆ On ∧ Tr 𝑥) ↔ (𝑥 ⊆ On ∧ ∀𝑦 ∈ 𝑥 Pred( E , On, 𝑦) ⊆ 𝑥))
25	6, 16, 24	3bitri 296	. . . . . . . . . . 11 ⊢ (𝑥 ∈ On ↔ (𝑥 ⊆ On ∧ ∀𝑦 ∈ 𝑥 Pred( E , On, 𝑦) ⊆ 𝑥))
26	25	anbi1i 622	. . . . . . . . . 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 6388	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ On)
28	27, 19	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → Pred( E , On, 𝑦) = 𝑦)
29	28	reseq2d 5980	. . . . . . . . . . . . . . 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 22	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ 𝑥 → 𝑦 ∈ 𝑥)
32		vex 3476	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑓 ∈ V
33	32	resex 6028	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 ↾ 𝑦) ∈ V
34	33	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ 𝑥 → (𝑓 ↾ 𝑦) ∈ V)
35	31, 34	opco2 8112	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ 𝑥 → (𝑦(𝐹 ∘ 2nd )(𝑓 ↾ 𝑦)) = (𝐹‘(𝑓 ↾ 𝑦)))
36	35	adantl 480	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → (𝑦(𝐹 ∘ 2nd )(𝑓 ↾ 𝑦)) = (𝐹‘(𝑓 ↾ 𝑦)))
37	30, 36	eqtrd 2770	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → (𝑦(𝐹 ∘ 2nd )(𝑓 ↾ Pred( E , On, 𝑦))) = (𝐹‘(𝑓 ↾ 𝑦)))
38	37	eqeq2d 2741	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → ((𝑓‘𝑦) = (𝑦(𝐹 ∘ 2nd )(𝑓 ↾ Pred( E , On, 𝑦))) ↔ (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦))))
39	38	ralbidva 3173	. . . . . . . . . . 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 573	. . . . . . . . . 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 276	. . . . . . . . 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 621	. . . . . . . 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 641	. . . . . . . 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 296	. . . . . . 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 1848	. . . . . 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 3069	. . . . . 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 277	. . . . 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 2800	. . . 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 4920	. . 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 2758	. 2 ⊢ frecs( E , On, (𝐹 ∘ 2nd )) = ∪ {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))}
51	1, 2, 50	3eqtri 2762	1 ⊢ recs(𝐹) = ∪ {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))}

Syntax hints:   ↔ wb 205   ∧ wa 394   ∧ w3a 1085   = wceq 1539  ∃wex 1779   ∈ wcel 2104  {cab 2707  ∀wral 3059  ∃wrex 3068  Vcvv 3472   ⊆ wss 3947  ∪ cuni 4907  Tr wtr 5264   E cep 5578   We wwe 5629   ↾ cres 5677   ∘ ccom 5679  Predcpred 6298  Ord word 6362  Oncon0 6363   Fn wfn 6537  ‘cfv 6542  (class class class)co 7411  2^nd c2nd 7976  frecscfrecs 8267  wrecscwrecs 8298  recscrecs 8372

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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7727

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6299  df-ord 6366  df-on 6367  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-fo 6548  df-fv 6550  df-ov 7414  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373

This theorem is referenced by:  recsfval  8383  tfrlem9  8387  dfrdg2  35071  dfrecs2  35226