Theorem dfrecs3 8307

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 8306	. 2 ⊢ recs(𝐹) = wrecs( E , On, 𝐹)
2		df-wrecs 8257	. 2 ⊢ wrecs( E , On, 𝐹) = frecs( E , On, (𝐹 ∘ 2nd ))
3		df-frecs 8226	. . 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 1095	. . . . . . . 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 3434	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ V
6	5	elon 6328	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ On ↔ Ord 𝑥)
7		ordsson 7732	. . . . . . . . . . . . . 14 ⊢ (Ord 𝑥 → 𝑥 ⊆ On)
8		ordtr 6333	. . . . . . . . . . . . . 14 ⊢ (Ord 𝑥 → Tr 𝑥)
9	7, 8	jca 511	. . . . . . . . . . . . 13 ⊢ (Ord 𝑥 → (𝑥 ⊆ On ∧ Tr 𝑥))
10		epweon 7724	. . . . . . . . . . . . . . . 16 ⊢ E We On
11		wess 5612	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ⊆ On → ( E We On → E We 𝑥))
12	10, 11	mpi 20	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ⊆ On → E We 𝑥)
13	12	anim1ci 617	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ⊆ On ∧ Tr 𝑥) → (Tr 𝑥 ∧ E We 𝑥))
14		df-ord 6322	. . . . . . . . . . . . . 14 ⊢ (Ord 𝑥 ↔ (Tr 𝑥 ∧ E We 𝑥))
15	13, 14	sylibr 234	. . . . . . . . . . . . 13 ⊢ ((𝑥 ⊆ On ∧ Tr 𝑥) → Ord 𝑥)
16	9, 15	impbii 209	. . . . . . . . . . . 12 ⊢ (Ord 𝑥 ↔ (𝑥 ⊆ On ∧ Tr 𝑥))
17		dftr3 5198	. . . . . . . . . . . . . 14 ⊢ (Tr 𝑥 ↔ ∀𝑦 ∈ 𝑥 𝑦 ⊆ 𝑥)
18		ssel2 3917	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ⊆ On ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ On)
19		predon 7735	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ On → Pred( E , On, 𝑦) = 𝑦)
20	19	sseq1d 3954	. . . . . . . . . . . . . . . 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 3159	. . . . . . . . . . . . . 14 ⊢ (𝑥 ⊆ On → (∀𝑦 ∈ 𝑥 Pred( E , On, 𝑦) ⊆ 𝑥 ↔ ∀𝑦 ∈ 𝑥 𝑦 ⊆ 𝑥))
23	17, 22	bitr4id 290	. . . . . . . . . . . . 13 ⊢ (𝑥 ⊆ On → (Tr 𝑥 ↔ ∀𝑦 ∈ 𝑥 Pred( E , On, 𝑦) ⊆ 𝑥))
24	23	pm5.32i 574	. . . . . . . . . . . 12 ⊢ ((𝑥 ⊆ On ∧ Tr 𝑥) ↔ (𝑥 ⊆ On ∧ ∀𝑦 ∈ 𝑥 Pred( E , On, 𝑦) ⊆ 𝑥))
25	6, 16, 24	3bitri 297	. . . . . . . . . . 11 ⊢ (𝑥 ∈ On ↔ (𝑥 ⊆ On ∧ ∀𝑦 ∈ 𝑥 Pred( E , On, 𝑦) ⊆ 𝑥))
26	25	anbi1i 625	. . . . . . . . . 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 6344	. . . . . . . . . . . . . . . . 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 5940	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → (𝑓 ↾ Pred( E , On, 𝑦)) = (𝑓 ↾ 𝑦))
30	29	oveq2d 7378	. . . . . . . . . . . . . 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 3434	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑓 ∈ V
33	32	resex 5990	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 ↾ 𝑦) ∈ V
34	33	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ 𝑥 → (𝑓 ↾ 𝑦) ∈ V)
35	31, 34	opco2 8069	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ 𝑥 → (𝑦(𝐹 ∘ 2nd )(𝑓 ↾ 𝑦)) = (𝐹‘(𝑓 ↾ 𝑦)))
36	35	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → (𝑦(𝐹 ∘ 2nd )(𝑓 ↾ 𝑦)) = (𝐹‘(𝑓 ↾ 𝑦)))
37	30, 36	eqtrd 2772	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → (𝑦(𝐹 ∘ 2nd )(𝑓 ↾ Pred( E , On, 𝑦))) = (𝐹‘(𝑓 ↾ 𝑦)))
38	37	eqeq2d 2748	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → ((𝑓‘𝑦) = (𝑦(𝐹 ∘ 2nd )(𝑓 ↾ Pred( E , On, 𝑦))) ↔ (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦))))
39	38	ralbidva 3159	. . . . . . . . . . 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 574	. . . . . . . . . 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 277	. . . . . . . . 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 624	. . . . . . . 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 646	. . . . . . . 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 297	. . . . . . 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 1850	. . . . . 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 3063	. . . . . 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 278	. . . . 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 2804	. . . 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 4863	. . 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 2760	. 2 ⊢ frecs( E , On, (𝐹 ∘ 2nd )) = ∪ {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))}
51	1, 2, 50	3eqtri 2764	1 ⊢ recs(𝐹) = ∪ {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))}

Syntax hints:   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542  ∃wex 1781   ∈ wcel 2114  {cab 2715  ∀wral 3052  ∃wrex 3062  Vcvv 3430   ⊆ wss 3890  ∪ cuni 4851  Tr wtr 5193   E cep 5525   We wwe 5578   ↾ cres 5628   ∘ ccom 5630  Predcpred 6260  Ord word 6318  Oncon0 6319   Fn wfn 6489  ‘cfv 6494  (class class class)co 7362  2^nd c2nd 7936  frecscfrecs 8225  wrecscwrecs 8256  recscrecs 8305

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5232  ax-nul 5242  ax-pr 5372  ax-un 7684

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5521  df-eprel 5526  df-po 5534  df-so 5535  df-fr 5579  df-we 5581  df-xp 5632  df-rel 5633  df-cnv 5634  df-co 5635  df-dm 5636  df-rn 5637  df-res 5638  df-ima 5639  df-pred 6261  df-ord 6322  df-on 6323  df-iota 6450  df-fun 6496  df-fn 6497  df-f 6498  df-fo 6500  df-fv 6502  df-ov 7365  df-2nd 7938  df-frecs 8226  df-wrecs 8257  df-recs 8306

This theorem is referenced by:  recsfval  8315  tfrlem9  8319  dfrdg2  35995  dfrecs2  36152