Theorem tfrlem11 7484

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 5791	. 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 7483	. . . . . . . 8 ⊢ (dom recs(𝐹) ∈ On → 𝐶 Fn suc dom recs(𝐹))
5		fnfun 5988	. . . . . . . 8 ⊢ (𝐶 Fn suc dom recs(𝐹) → Fun 𝐶)
6	4, 5	syl 17	. . . . . . 7 ⊢ (dom recs(𝐹) ∈ On → Fun 𝐶)
7		ssun1 3776	. . . . . . . . 9 ⊢ recs(𝐹) ⊆ (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩})
8	7, 3	sseqtr4i 3638	. . . . . . . 8 ⊢ recs(𝐹) ⊆ 𝐶
9	2	tfrlem9 7481	. . . . . . . . 9 ⊢ (𝐵 ∈ dom recs(𝐹) → (recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵)))
10		funssfv 6209	. . . . . . . . . . . 12 ⊢ ((Fun 𝐶 ∧ recs(𝐹) ⊆ 𝐶 ∧ 𝐵 ∈ dom recs(𝐹)) → (𝐶‘𝐵) = (recs(𝐹)‘𝐵))
11	10	3expa 1265	. . . . . . . . . . 11 ⊢ (((Fun 𝐶 ∧ recs(𝐹) ⊆ 𝐶) ∧ 𝐵 ∈ dom recs(𝐹)) → (𝐶‘𝐵) = (recs(𝐹)‘𝐵))
12	11	adantrl 752	. . . . . . . . . 10 ⊢ (((Fun 𝐶 ∧ recs(𝐹) ⊆ 𝐶) ∧ (dom recs(𝐹) ∈ On ∧ 𝐵 ∈ dom recs(𝐹))) → (𝐶‘𝐵) = (recs(𝐹)‘𝐵))
13		onelss 5766	. . . . . . . . . . . 12 ⊢ (dom recs(𝐹) ∈ On → (𝐵 ∈ dom recs(𝐹) → 𝐵 ⊆ dom recs(𝐹)))
14	13	imp 445	. . . . . . . . . . 11 ⊢ ((dom recs(𝐹) ∈ On ∧ 𝐵 ∈ dom recs(𝐹)) → 𝐵 ⊆ dom recs(𝐹))
15		fun2ssres 5931	. . . . . . . . . . . . 13 ⊢ ((Fun 𝐶 ∧ recs(𝐹) ⊆ 𝐶 ∧ 𝐵 ⊆ dom recs(𝐹)) → (𝐶 ↾ 𝐵) = (recs(𝐹) ↾ 𝐵))
16	15	3expa 1265	. . . . . . . . . . . 12 ⊢ (((Fun 𝐶 ∧ recs(𝐹) ⊆ 𝐶) ∧ 𝐵 ⊆ dom recs(𝐹)) → (𝐶 ↾ 𝐵) = (recs(𝐹) ↾ 𝐵))
17	16	fveq2d 6195	. . . . . . . . . . 11 ⊢ (((Fun 𝐶 ∧ recs(𝐹) ⊆ 𝐶) ∧ 𝐵 ⊆ dom recs(𝐹)) → (𝐹‘(𝐶 ↾ 𝐵)) = (𝐹‘(recs(𝐹) ↾ 𝐵)))
18	14, 17	sylan2 491	. . . . . . . . . 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 2637	. . . . . . . . 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	syl5ibr 236	. . . . . . . 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 717	. . . . . . 7 ⊢ ((Fun 𝐶 ∧ (dom recs(𝐹) ∈ On ∧ 𝐵 ∈ dom recs(𝐹))) → (𝐵 ∈ dom recs(𝐹) → (𝐶‘𝐵) = (𝐹‘(𝐶 ↾ 𝐵))))
22	6, 21	sylan 488	. . . . . 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 631	. . . . 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 4932	. . . . . . . . 9 ⊢ ⟨𝐵, (𝐹‘(𝐶 ↾ 𝐵))⟩ ∈ V
27	26	snid 4208	. . . . . . . 8 ⊢ ⟨𝐵, (𝐹‘(𝐶 ↾ 𝐵))⟩ ∈ {⟨𝐵, (𝐹‘(𝐶 ↾ 𝐵))⟩}
28		opeq1 4402	. . . . . . . . . . 11 ⊢ (𝐵 = dom recs(𝐹) → ⟨𝐵, (𝐹‘(𝐶 ↾ 𝐵))⟩ = ⟨dom recs(𝐹), (𝐹‘(𝐶 ↾ 𝐵))⟩)
29	28	adantl 482	. . . . . . . . . 10 ⊢ ((dom recs(𝐹) ∈ On ∧ 𝐵 = dom recs(𝐹)) → ⟨𝐵, (𝐹‘(𝐶 ↾ 𝐵))⟩ = ⟨dom recs(𝐹), (𝐹‘(𝐶 ↾ 𝐵))⟩)
30		eqimss 3657	. . . . . . . . . . . . . 14 ⊢ (𝐵 = dom recs(𝐹) → 𝐵 ⊆ dom recs(𝐹))
31	8, 15	mp3an2 1412	. . . . . . . . . . . . . 14 ⊢ ((Fun 𝐶 ∧ 𝐵 ⊆ dom recs(𝐹)) → (𝐶 ↾ 𝐵) = (recs(𝐹) ↾ 𝐵))
32	6, 30, 31	syl2an 494	. . . . . . . . . . . . 13 ⊢ ((dom recs(𝐹) ∈ On ∧ 𝐵 = dom recs(𝐹)) → (𝐶 ↾ 𝐵) = (recs(𝐹) ↾ 𝐵))
33		reseq2 5391	. . . . . . . . . . . . . . 15 ⊢ (𝐵 = dom recs(𝐹) → (recs(𝐹) ↾ 𝐵) = (recs(𝐹) ↾ dom recs(𝐹)))
34	2	tfrlem6 7478	. . . . . . . . . . . . . . . 16 ⊢ Rel recs(𝐹)
35		resdm 5441	. . . . . . . . . . . . . . . 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	syl6eq 2672	. . . . . . . . . . . . . 14 ⊢ (𝐵 = dom recs(𝐹) → (recs(𝐹) ↾ 𝐵) = recs(𝐹))
38	37	adantl 482	. . . . . . . . . . . . 13 ⊢ ((dom recs(𝐹) ∈ On ∧ 𝐵 = dom recs(𝐹)) → (recs(𝐹) ↾ 𝐵) = recs(𝐹))
39	32, 38	eqtrd 2656	. . . . . . . . . . . 12 ⊢ ((dom recs(𝐹) ∈ On ∧ 𝐵 = dom recs(𝐹)) → (𝐶 ↾ 𝐵) = recs(𝐹))
40	39	fveq2d 6195	. . . . . . . . . . 11 ⊢ ((dom recs(𝐹) ∈ On ∧ 𝐵 = dom recs(𝐹)) → (𝐹‘(𝐶 ↾ 𝐵)) = (𝐹‘recs(𝐹)))
41	40	opeq2d 4409	. . . . . . . . . 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 2656	. . . . . . . . 9 ⊢ ((dom recs(𝐹) ∈ On ∧ 𝐵 = dom recs(𝐹)) → ⟨𝐵, (𝐹‘(𝐶 ↾ 𝐵))⟩ = ⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩)
43	42	sneqd 4189	. . . . . . . 8 ⊢ ((dom recs(𝐹) ∈ On ∧ 𝐵 = dom recs(𝐹)) → {⟨𝐵, (𝐹‘(𝐶 ↾ 𝐵))⟩} = {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩})
44	27, 43	syl5eleq 2707	. . . . . . 7 ⊢ ((dom recs(𝐹) ∈ On ∧ 𝐵 = dom recs(𝐹)) → ⟨𝐵, (𝐹‘(𝐶 ↾ 𝐵))⟩ ∈ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩})
45		elun2 3781	. . . . . . 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	syl6eleqr 2712	. . . . 5 ⊢ ((dom recs(𝐹) ∈ On ∧ 𝐵 = dom recs(𝐹)) → ⟨𝐵, (𝐹‘(𝐶 ↾ 𝐵))⟩ ∈ 𝐶)
48	4	adantr 481	. . . . . 6 ⊢ ((dom recs(𝐹) ∈ On ∧ 𝐵 = dom recs(𝐹)) → 𝐶 Fn suc dom recs(𝐹))
49		simpr 477	. . . . . . 7 ⊢ ((dom recs(𝐹) ∈ On ∧ 𝐵 = dom recs(𝐹)) → 𝐵 = dom recs(𝐹))
50		sucidg 5803	. . . . . . . 8 ⊢ (dom recs(𝐹) ∈ On → dom recs(𝐹) ∈ suc dom recs(𝐹))
51	50	adantr 481	. . . . . . 7 ⊢ ((dom recs(𝐹) ∈ On ∧ 𝐵 = dom recs(𝐹)) → dom recs(𝐹) ∈ suc dom recs(𝐹))
52	49, 51	eqeltrd 2701	. . . . . 6 ⊢ ((dom recs(𝐹) ∈ On ∧ 𝐵 = dom recs(𝐹)) → 𝐵 ∈ suc dom recs(𝐹))
53		fnopfvb 6237	. . . . . 6 ⊢ ((𝐶 Fn suc dom recs(𝐹) ∧ 𝐵 ∈ suc dom recs(𝐹)) → ((𝐶‘𝐵) = (𝐹‘(𝐶 ↾ 𝐵)) ↔ ⟨𝐵, (𝐹‘(𝐶 ↾ 𝐵))⟩ ∈ 𝐶))
54	48, 52, 53	syl2anc 693	. . . . 5 ⊢ ((dom recs(𝐹) ∈ On ∧ 𝐵 = dom recs(𝐹)) → ((𝐶‘𝐵) = (𝐹‘(𝐶 ↾ 𝐵)) ↔ ⟨𝐵, (𝐹‘(𝐶 ↾ 𝐵))⟩ ∈ 𝐶))
55	47, 54	mpbird 247	. . . 4 ⊢ ((dom recs(𝐹) ∈ On ∧ 𝐵 = dom recs(𝐹)) → (𝐶‘𝐵) = (𝐹‘(𝐶 ↾ 𝐵)))
56	55	ex 450	. . 3 ⊢ (dom recs(𝐹) ∈ On → (𝐵 = dom recs(𝐹) → (𝐶‘𝐵) = (𝐹‘(𝐶 ↾ 𝐵))))
57	25, 56	jaod 395	. 2 ⊢ (dom recs(𝐹) ∈ On → ((𝐵 ∈ dom recs(𝐹) ∨ 𝐵 = dom recs(𝐹)) → (𝐶‘𝐵) = (𝐹‘(𝐶 ↾ 𝐵))))
58	1, 57	syl5 34	1 ⊢ (dom recs(𝐹) ∈ On → (𝐵 ∈ suc dom recs(𝐹) → (𝐶‘𝐵) = (𝐹‘(𝐶 ↾ 𝐵))))

Syntax hints:   → wi 4   ↔ wb 196   ∨ wo 383   ∧ wa 384   = wceq 1483   ∈ wcel 1990  {cab 2608  ∀wral 2912  ∃wrex 2913   ∪ cun 3572   ⊆ wss 3574  {csn 4177  ⟨cop 4183  dom cdm 5114   ↾ cres 5116  Rel wrel 5119  Oncon0 5723  suc csuc 5725  Fun wfun 5882   Fn wfn 5883  ‘cfv 5888  recscrecs 7467

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-fv 5896  df-wrecs 7407  df-recs 7468

This theorem is referenced by:  tfrlem12  7485