Theorem dfrecs3 8364

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 8363	. 2 ⊢ recs(𝐹) = wrecs( E , On, 𝐹)
2		df-wrecs 8314	. 2 ⊢ wrecs( E , On, 𝐹) = frecs( E , On, (𝐹 ∘ 2nd ))
3		df-frecs 8283	. . 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 1111	. . . . . . . 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 3457	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ V
6	5	elon 6370	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ On ↔ Ord 𝑥)
7		ordsson 7785	. . . . . . . . . . . . . 14 ⊢ (Ord 𝑥 → 𝑥 ⊆ On)
8		ordtr 6375	. . . . . . . . . . . . . 14 ⊢ (Ord 𝑥 → Tr 𝑥)
9	7, 8	jca 521	. . . . . . . . . . . . 13 ⊢ (Ord 𝑥 → (𝑥 ⊆ On ∧ Tr 𝑥))
10		epweon 7777	. . . . . . . . . . . . . . . 16 ⊢ E We On
11		wess 5645	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ⊆ On → ( E We On → E We 𝑥))
12	10, 11	mpi 21	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ⊆ On → E We 𝑥)
13	12	anim1ci 628	. . . . . . . . . . . . . 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 5221	. . . . . . . . . . . . . 14 ⊢ (Tr 𝑥 ↔ ∀𝑦 ∈ 𝑥 𝑦 ⊆ 𝑥)
18		ssel2 3929	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ⊆ On ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ On)
19		predon 7788	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ On → Pred( E , On, 𝑦) = 𝑦)
20	19	sseq1d 3965	. . . . . . . . . . . . . . . 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 3185	. . . . . . . . . . . . . 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 585	. . . . . . . . . . . 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 636	. . . . . . . . . 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 5976	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → (𝑓 ↾ Pred( E , On, 𝑦)) = (𝑓 ↾ 𝑦))
30	29	oveq2d 7432	. . . . . . . . . . . . . 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 3457	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑓 ∈ V
33	32	resex 6026	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 ↾ 𝑦) ∈ V
34	33	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ 𝑥 → (𝑓 ↾ 𝑦) ∈ V)
35	31, 34	opco2 8124	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ 𝑥 → (𝑦(𝐹 ∘ 2nd )(𝑓 ↾ 𝑦)) = (𝐹‘(𝑓 ↾ 𝑦)))
36	35	adantl 487	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → (𝑦(𝐹 ∘ 2nd )(𝑓 ↾ 𝑦)) = (𝐹‘(𝑓 ↾ 𝑦)))
37	30, 36	eqtrd 2797	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → (𝑦(𝐹 ∘ 2nd )(𝑓 ↾ Pred( E , On, 𝑦))) = (𝐹‘(𝑓 ↾ 𝑦)))
38	37	eqeq2d 2773	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → ((𝑓‘𝑦) = (𝑦(𝐹 ∘ 2nd )(𝑓 ↾ Pred( E , On, 𝑦))) ↔ (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦))))
39	38	ralbidva 3185	. . . . . . . . . . 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 585	. . . . . . . . . 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 635	. . . . . . . 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 658	. . . . . . . 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 1881	. . . . . 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 3089	. . . . . 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 2829	. . . 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 4882	. . 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 2785	. 2 ⊢ frecs( E , On, (𝐹 ∘ 2nd )) = ∪ {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))}
51	1, 2, 50	3eqtri 2789	1 ⊢ recs(𝐹) = ∪ {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))}

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570  ∃wex 1812   ∈ wcel 2145  {cab 2740  ∀wral 3078  ∃wrex 3088  Vcvv 3453   ⊆ wss 3902  ∪ cuni 4870  Tr wtr 5216   E cep 5558   We wwe 5611   ↾ cres 5661   ∘ ccom 5663  Predcpred 6302  Ord word 6360  Oncon0 6361   Fn wfn 6532  ‘cfv 6537  (class class class)co 7416  2^nd c2nd 7988  frecscfrecs 8282  wrecscwrecs 8313  recscrecs 8362

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pr 5402  ax-un 7739

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  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 7419  df-2nd 7990  df-frecs 8283  df-wrecs 8314  df-recs 8363

This theorem is used by:  recsfval  8372  tfrlem9  8377  dfrdg2  36359  dfrecs2  36516