Theorem tfrlem11 8359

Description: Lemma for transfinite recursion. Compute the value of 𝐶. (Contributed by NM, 18-Aug-1994.) (Revised by Mario Carneiro, 9-May-2015.)

Hypotheses

Ref	Expression
tfrlem.1	⊢ 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))}
tfrlem.3	⊢ 𝐶 = (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩})

Assertion

Ref	Expression
tfrlem11	⊢ (dom recs(𝐹) ∈ On → (𝐵 ∈ suc dom recs(𝐹) → (𝐶‘𝐵) = (𝐹‘(𝐶 ↾ 𝐵))))

Distinct variable groups:   𝑥,𝑓,𝑦,𝐵   𝐶,𝑓,𝑥,𝑦   𝑓,𝐹,𝑥,𝑦

Allowed substitution hints:   𝐴(𝑥,𝑦,𝑓)

Proof of Theorem tfrlem11

Step	Hyp	Ref	Expression
1		elsuci 6404	. 2 ⊢ (𝐵 ∈ suc dom recs(𝐹) → (𝐵 ∈ dom recs(𝐹) ∨ 𝐵 = dom recs(𝐹)))
2		tfrlem.1	. . . . . . . . 9 ⊢ 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))}
3		tfrlem.3	. . . . . . . . 9 ⊢ 𝐶 = (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩})
4	2, 3	tfrlem10 8358	. . . . . . . 8 ⊢ (dom recs(𝐹) ∈ On → 𝐶 Fn suc dom recs(𝐹))
5		fnfun 6621	. . . . . . . 8 ⊢ (𝐶 Fn suc dom recs(𝐹) → Fun 𝐶)
6	4, 5	syl 17	. . . . . . 7 ⊢ (dom recs(𝐹) ∈ On → Fun 𝐶)
7		ssun1 4144	. . . . . . . . 9 ⊢ recs(𝐹) ⊆ (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩})
8	7, 3	sseqtrri 3999	. . . . . . . 8 ⊢ recs(𝐹) ⊆ 𝐶
9	2	tfrlem9 8356	. . . . . . . . 9 ⊢ (𝐵 ∈ dom recs(𝐹) → (recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵)))
10		funssfv 6882	. . . . . . . . . . . 12 ⊢ ((Fun 𝐶 ∧ recs(𝐹) ⊆ 𝐶 ∧ 𝐵 ∈ dom recs(𝐹)) → (𝐶‘𝐵) = (recs(𝐹)‘𝐵))
11	10	3expa 1118	. . . . . . . . . . 11 ⊢ (((Fun 𝐶 ∧ recs(𝐹) ⊆ 𝐶) ∧ 𝐵 ∈ dom recs(𝐹)) → (𝐶‘𝐵) = (recs(𝐹)‘𝐵))
12	11	adantrl 716	. . . . . . . . . 10 ⊢ (((Fun 𝐶 ∧ recs(𝐹) ⊆ 𝐶) ∧ (dom recs(𝐹) ∈ On ∧ 𝐵 ∈ dom recs(𝐹))) → (𝐶‘𝐵) = (recs(𝐹)‘𝐵))
13		onelss 6377	. . . . . . . . . . . 12 ⊢ (dom recs(𝐹) ∈ On → (𝐵 ∈ dom recs(𝐹) → 𝐵 ⊆ dom recs(𝐹)))
14	13	imp 406	. . . . . . . . . . 11 ⊢ ((dom recs(𝐹) ∈ On ∧ 𝐵 ∈ dom recs(𝐹)) → 𝐵 ⊆ dom recs(𝐹))
15		fun2ssres 6564	. . . . . . . . . . . . 13 ⊢ ((Fun 𝐶 ∧ recs(𝐹) ⊆ 𝐶 ∧ 𝐵 ⊆ dom recs(𝐹)) → (𝐶 ↾ 𝐵) = (recs(𝐹) ↾ 𝐵))
16	15	3expa 1118	. . . . . . . . . . . 12 ⊢ (((Fun 𝐶 ∧ recs(𝐹) ⊆ 𝐶) ∧ 𝐵 ⊆ dom recs(𝐹)) → (𝐶 ↾ 𝐵) = (recs(𝐹) ↾ 𝐵))
17	16	fveq2d 6865	. . . . . . . . . . 11 ⊢ (((Fun 𝐶 ∧ recs(𝐹) ⊆ 𝐶) ∧ 𝐵 ⊆ dom recs(𝐹)) → (𝐹‘(𝐶 ↾ 𝐵)) = (𝐹‘(recs(𝐹) ↾ 𝐵)))
18	14, 17	sylan2 593	. . . . . . . . . 10 ⊢ (((Fun 𝐶 ∧ recs(𝐹) ⊆ 𝐶) ∧ (dom recs(𝐹) ∈ On ∧ 𝐵 ∈ dom recs(𝐹))) → (𝐹‘(𝐶 ↾ 𝐵)) = (𝐹‘(recs(𝐹) ↾ 𝐵)))
19	12, 18	eqeq12d 2746	. . . . . . . . 9 ⊢ (((Fun 𝐶 ∧ recs(𝐹) ⊆ 𝐶) ∧ (dom recs(𝐹) ∈ On ∧ 𝐵 ∈ dom recs(𝐹))) → ((𝐶‘𝐵) = (𝐹‘(𝐶 ↾ 𝐵)) ↔ (recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵))))
20	9, 19	imbitrrid 246	. . . . . . . 8 ⊢ (((Fun 𝐶 ∧ recs(𝐹) ⊆ 𝐶) ∧ (dom recs(𝐹) ∈ On ∧ 𝐵 ∈ dom recs(𝐹))) → (𝐵 ∈ dom recs(𝐹) → (𝐶‘𝐵) = (𝐹‘(𝐶 ↾ 𝐵))))
21	8, 20	mpanl2 701	. . . . . . 7 ⊢ ((Fun 𝐶 ∧ (dom recs(𝐹) ∈ On ∧ 𝐵 ∈ dom recs(𝐹))) → (𝐵 ∈ dom recs(𝐹) → (𝐶‘𝐵) = (𝐹‘(𝐶 ↾ 𝐵))))
22	6, 21	sylan 580	. . . . . 6 ⊢ ((dom recs(𝐹) ∈ On ∧ (dom recs(𝐹) ∈ On ∧ 𝐵 ∈ dom recs(𝐹))) → (𝐵 ∈ dom recs(𝐹) → (𝐶‘𝐵) = (𝐹‘(𝐶 ↾ 𝐵))))
23	22	exp32 420	. . . . 5 ⊢ (dom recs(𝐹) ∈ On → (dom recs(𝐹) ∈ On → (𝐵 ∈ dom recs(𝐹) → (𝐵 ∈ dom recs(𝐹) → (𝐶‘𝐵) = (𝐹‘(𝐶 ↾ 𝐵))))))
24	23	pm2.43i 52	. . . 4 ⊢ (dom recs(𝐹) ∈ On → (𝐵 ∈ dom recs(𝐹) → (𝐵 ∈ dom recs(𝐹) → (𝐶‘𝐵) = (𝐹‘(𝐶 ↾ 𝐵)))))
25	24	pm2.43d 53	. . 3 ⊢ (dom recs(𝐹) ∈ On → (𝐵 ∈ dom recs(𝐹) → (𝐶‘𝐵) = (𝐹‘(𝐶 ↾ 𝐵))))
26		opex 5427	. . . . . . . . 9 ⊢ ⟨𝐵, (𝐹‘(𝐶 ↾ 𝐵))⟩ ∈ V
27	26	snid 4629	. . . . . . . 8 ⊢ ⟨𝐵, (𝐹‘(𝐶 ↾ 𝐵))⟩ ∈ {⟨𝐵, (𝐹‘(𝐶 ↾ 𝐵))⟩}
28		opeq1 4840	. . . . . . . . . . 11 ⊢ (𝐵 = dom recs(𝐹) → ⟨𝐵, (𝐹‘(𝐶 ↾ 𝐵))⟩ = ⟨dom recs(𝐹), (𝐹‘(𝐶 ↾ 𝐵))⟩)
29	28	adantl 481	. . . . . . . . . 10 ⊢ ((dom recs(𝐹) ∈ On ∧ 𝐵 = dom recs(𝐹)) → ⟨𝐵, (𝐹‘(𝐶 ↾ 𝐵))⟩ = ⟨dom recs(𝐹), (𝐹‘(𝐶 ↾ 𝐵))⟩)
30		eqimss 4008	. . . . . . . . . . . . . 14 ⊢ (𝐵 = dom recs(𝐹) → 𝐵 ⊆ dom recs(𝐹))
31	8, 15	mp3an2 1451	. . . . . . . . . . . . . 14 ⊢ ((Fun 𝐶 ∧ 𝐵 ⊆ dom recs(𝐹)) → (𝐶 ↾ 𝐵) = (recs(𝐹) ↾ 𝐵))
32	6, 30, 31	syl2an 596	. . . . . . . . . . . . 13 ⊢ ((dom recs(𝐹) ∈ On ∧ 𝐵 = dom recs(𝐹)) → (𝐶 ↾ 𝐵) = (recs(𝐹) ↾ 𝐵))
33		reseq2 5948	. . . . . . . . . . . . . . 15 ⊢ (𝐵 = dom recs(𝐹) → (recs(𝐹) ↾ 𝐵) = (recs(𝐹) ↾ dom recs(𝐹)))
34	2	tfrlem6 8353	. . . . . . . . . . . . . . . 16 ⊢ Rel recs(𝐹)
35		resdm 6000	. . . . . . . . . . . . . . . 16 ⊢ (Rel recs(𝐹) → (recs(𝐹) ↾ dom recs(𝐹)) = recs(𝐹))
36	34, 35	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ (recs(𝐹) ↾ dom recs(𝐹)) = recs(𝐹)
37	33, 36	eqtrdi 2781	. . . . . . . . . . . . . 14 ⊢ (𝐵 = dom recs(𝐹) → (recs(𝐹) ↾ 𝐵) = recs(𝐹))
38	37	adantl 481	. . . . . . . . . . . . 13 ⊢ ((dom recs(𝐹) ∈ On ∧ 𝐵 = dom recs(𝐹)) → (recs(𝐹) ↾ 𝐵) = recs(𝐹))
39	32, 38	eqtrd 2765	. . . . . . . . . . . 12 ⊢ ((dom recs(𝐹) ∈ On ∧ 𝐵 = dom recs(𝐹)) → (𝐶 ↾ 𝐵) = recs(𝐹))
40	39	fveq2d 6865	. . . . . . . . . . 11 ⊢ ((dom recs(𝐹) ∈ On ∧ 𝐵 = dom recs(𝐹)) → (𝐹‘(𝐶 ↾ 𝐵)) = (𝐹‘recs(𝐹)))
41	40	opeq2d 4847	. . . . . . . . . 10 ⊢ ((dom recs(𝐹) ∈ On ∧ 𝐵 = dom recs(𝐹)) → ⟨dom recs(𝐹), (𝐹‘(𝐶 ↾ 𝐵))⟩ = ⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩)
42	29, 41	eqtrd 2765	. . . . . . . . 9 ⊢ ((dom recs(𝐹) ∈ On ∧ 𝐵 = dom recs(𝐹)) → ⟨𝐵, (𝐹‘(𝐶 ↾ 𝐵))⟩ = ⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩)
43	42	sneqd 4604	. . . . . . . 8 ⊢ ((dom recs(𝐹) ∈ On ∧ 𝐵 = dom recs(𝐹)) → {⟨𝐵, (𝐹‘(𝐶 ↾ 𝐵))⟩} = {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩})
44	27, 43	eleqtrid 2835	. . . . . . 7 ⊢ ((dom recs(𝐹) ∈ On ∧ 𝐵 = dom recs(𝐹)) → ⟨𝐵, (𝐹‘(𝐶 ↾ 𝐵))⟩ ∈ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩})
45		elun2 4149	. . . . . . 7 ⊢ (⟨𝐵, (𝐹‘(𝐶 ↾ 𝐵))⟩ ∈ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩} → ⟨𝐵, (𝐹‘(𝐶 ↾ 𝐵))⟩ ∈ (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}))
46	44, 45	syl 17	. . . . . 6 ⊢ ((dom recs(𝐹) ∈ On ∧ 𝐵 = dom recs(𝐹)) → ⟨𝐵, (𝐹‘(𝐶 ↾ 𝐵))⟩ ∈ (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}))
47	46, 3	eleqtrrdi 2840	. . . . 5 ⊢ ((dom recs(𝐹) ∈ On ∧ 𝐵 = dom recs(𝐹)) → ⟨𝐵, (𝐹‘(𝐶 ↾ 𝐵))⟩ ∈ 𝐶)
48		simpr 484	. . . . . . 7 ⊢ ((dom recs(𝐹) ∈ On ∧ 𝐵 = dom recs(𝐹)) → 𝐵 = dom recs(𝐹))
49		sucidg 6418	. . . . . . . 8 ⊢ (dom recs(𝐹) ∈ On → dom recs(𝐹) ∈ suc dom recs(𝐹))
50	49	adantr 480	. . . . . . 7 ⊢ ((dom recs(𝐹) ∈ On ∧ 𝐵 = dom recs(𝐹)) → dom recs(𝐹) ∈ suc dom recs(𝐹))
51	48, 50	eqeltrd 2829	. . . . . 6 ⊢ ((dom recs(𝐹) ∈ On ∧ 𝐵 = dom recs(𝐹)) → 𝐵 ∈ suc dom recs(𝐹))
52		fnopfvb 6915	. . . . . 6 ⊢ ((𝐶 Fn suc dom recs(𝐹) ∧ 𝐵 ∈ suc dom recs(𝐹)) → ((𝐶‘𝐵) = (𝐹‘(𝐶 ↾ 𝐵)) ↔ ⟨𝐵, (𝐹‘(𝐶 ↾ 𝐵))⟩ ∈ 𝐶))
53	4, 51, 52	syl2an2r 685	. . . . 5 ⊢ ((dom recs(𝐹) ∈ On ∧ 𝐵 = dom recs(𝐹)) → ((𝐶‘𝐵) = (𝐹‘(𝐶 ↾ 𝐵)) ↔ ⟨𝐵, (𝐹‘(𝐶 ↾ 𝐵))⟩ ∈ 𝐶))
54	47, 53	mpbird 257	. . . 4 ⊢ ((dom recs(𝐹) ∈ On ∧ 𝐵 = dom recs(𝐹)) → (𝐶‘𝐵) = (𝐹‘(𝐶 ↾ 𝐵)))
55	54	ex 412	. . 3 ⊢ (dom recs(𝐹) ∈ On → (𝐵 = dom recs(𝐹) → (𝐶‘𝐵) = (𝐹‘(𝐶 ↾ 𝐵))))
56	25, 55	jaod 859	. 2 ⊢ (dom recs(𝐹) ∈ On → ((𝐵 ∈ dom recs(𝐹) ∨ 𝐵 = dom recs(𝐹)) → (𝐶‘𝐵) = (𝐹‘(𝐶 ↾ 𝐵))))
57	1, 56	syl5 34	1 ⊢ (dom recs(𝐹) ∈ On → (𝐵 ∈ suc dom recs(𝐹) → (𝐶‘𝐵) = (𝐹‘(𝐶 ↾ 𝐵))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   = wceq 1540   ∈ wcel 2109  {cab 2708  ∀wral 3045  ∃wrex 3054   ∪ cun 3915   ⊆ wss 3917  {csn 4592  ⟨cop 4598  dom cdm 5641   ↾ cres 5643  Rel wrel 5646  Oncon0 6335  suc csuc 6337  Fun wfun 6508   Fn wfn 6509  ‘cfv 6514  recscrecs 8342

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 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-fo 6520  df-fv 6522  df-ov 7393  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343

This theorem is referenced by:  tfrlem12  8360