Theorem frec0g 6628

Description: The initial value resulting from finite recursive definition generation. (Contributed by Jim Kingdon, 7-May-2020.)

Assertion

Ref	Expression
frec0g	⊢ (𝐴 ∈ 𝑉 → (frec(𝐹, 𝐴)‘∅) = 𝐴)

Proof of Theorem frec0g

Dummy variables 𝑔 𝑚 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dm0 4970	. . . . . . . . . 10 ⊢ dom ∅ = ∅
2	1	biantrur 303	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐴 ↔ (dom ∅ = ∅ ∧ 𝑥 ∈ 𝐴))
3		vex 2816	. . . . . . . . . . . . . . . 16 ⊢ 𝑚 ∈ V
4		nsuceq0g 4539	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 ∈ V → suc 𝑚 ≠ ∅)
5	3, 4	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ suc 𝑚 ≠ ∅
6	5	nesymi 2458	. . . . . . . . . . . . . 14 ⊢ ¬ ∅ = suc 𝑚
7	1	eqeq1i 2240	. . . . . . . . . . . . . 14 ⊢ (dom ∅ = suc 𝑚 ↔ ∅ = suc 𝑚)
8	6, 7	mtbir 678	. . . . . . . . . . . . 13 ⊢ ¬ dom ∅ = suc 𝑚
9	8	intnanr 938	. . . . . . . . . . . 12 ⊢ ¬ (dom ∅ = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(∅‘𝑚)))
10	9	a1i 9	. . . . . . . . . . 11 ⊢ (𝑚 ∈ ω → ¬ (dom ∅ = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(∅‘𝑚))))
11	10	nrex 2634	. . . . . . . . . 10 ⊢ ¬ ∃𝑚 ∈ ω (dom ∅ = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(∅‘𝑚)))
12	11	biorfi 754	. . . . . . . . 9 ⊢ ((dom ∅ = ∅ ∧ 𝑥 ∈ 𝐴) ↔ ((dom ∅ = ∅ ∧ 𝑥 ∈ 𝐴) ∨ ∃𝑚 ∈ ω (dom ∅ = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(∅‘𝑚)))))
13		orcom 736	. . . . . . . . 9 ⊢ (((dom ∅ = ∅ ∧ 𝑥 ∈ 𝐴) ∨ ∃𝑚 ∈ ω (dom ∅ = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(∅‘𝑚)))) ↔ (∃𝑚 ∈ ω (dom ∅ = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(∅‘𝑚))) ∨ (dom ∅ = ∅ ∧ 𝑥 ∈ 𝐴)))
14	2, 12, 13	3bitri 206	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐴 ↔ (∃𝑚 ∈ ω (dom ∅ = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(∅‘𝑚))) ∨ (dom ∅ = ∅ ∧ 𝑥 ∈ 𝐴)))
15	14	abbii 2348	. . . . . . 7 ⊢ {𝑥 ∣ 𝑥 ∈ 𝐴} = {𝑥 ∣ (∃𝑚 ∈ ω (dom ∅ = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(∅‘𝑚))) ∨ (dom ∅ = ∅ ∧ 𝑥 ∈ 𝐴))}
16		abid2 2355	. . . . . . 7 ⊢ {𝑥 ∣ 𝑥 ∈ 𝐴} = 𝐴
17	15, 16	eqtr3i 2255	. . . . . 6 ⊢ {𝑥 ∣ (∃𝑚 ∈ ω (dom ∅ = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(∅‘𝑚))) ∨ (dom ∅ = ∅ ∧ 𝑥 ∈ 𝐴))} = 𝐴
18		elex 2825	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
19	17, 18	eqeltrid 2319	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∣ (∃𝑚 ∈ ω (dom ∅ = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(∅‘𝑚))) ∨ (dom ∅ = ∅ ∧ 𝑥 ∈ 𝐴))} ∈ V)
20		0ex 4237	. . . . . . 7 ⊢ ∅ ∈ V
21		dmeq 4956	. . . . . . . . . . . . 13 ⊢ (𝑔 = ∅ → dom 𝑔 = dom ∅)
22	21	eqeq1d 2241	. . . . . . . . . . . 12 ⊢ (𝑔 = ∅ → (dom 𝑔 = suc 𝑚 ↔ dom ∅ = suc 𝑚))
23		fveq1 5669	. . . . . . . . . . . . . 14 ⊢ (𝑔 = ∅ → (𝑔‘𝑚) = (∅‘𝑚))
24	23	fveq2d 5674	. . . . . . . . . . . . 13 ⊢ (𝑔 = ∅ → (𝐹‘(𝑔‘𝑚)) = (𝐹‘(∅‘𝑚)))
25	24	eleq2d 2302	. . . . . . . . . . . 12 ⊢ (𝑔 = ∅ → (𝑥 ∈ (𝐹‘(𝑔‘𝑚)) ↔ 𝑥 ∈ (𝐹‘(∅‘𝑚))))
26	22, 25	anbi12d 473	. . . . . . . . . . 11 ⊢ (𝑔 = ∅ → ((dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ↔ (dom ∅ = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(∅‘𝑚)))))
27	26	rexbidv 2543	. . . . . . . . . 10 ⊢ (𝑔 = ∅ → (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ↔ ∃𝑚 ∈ ω (dom ∅ = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(∅‘𝑚)))))
28	21	eqeq1d 2241	. . . . . . . . . . 11 ⊢ (𝑔 = ∅ → (dom 𝑔 = ∅ ↔ dom ∅ = ∅))
29	28	anbi1d 465	. . . . . . . . . 10 ⊢ (𝑔 = ∅ → ((dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐴) ↔ (dom ∅ = ∅ ∧ 𝑥 ∈ 𝐴)))
30	27, 29	orbi12d 801	. . . . . . . . 9 ⊢ (𝑔 = ∅ → ((∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐴)) ↔ (∃𝑚 ∈ ω (dom ∅ = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(∅‘𝑚))) ∨ (dom ∅ = ∅ ∧ 𝑥 ∈ 𝐴))))
31	30	abbidv 2352	. . . . . . . 8 ⊢ (𝑔 = ∅ → {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐴))} = {𝑥 ∣ (∃𝑚 ∈ ω (dom ∅ = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(∅‘𝑚))) ∨ (dom ∅ = ∅ ∧ 𝑥 ∈ 𝐴))})
32		eqid 2232	. . . . . . . 8 ⊢ (𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐴))}) = (𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐴))})
33	31, 32	fvmptg 5753	. . . . . . 7 ⊢ ((∅ ∈ V ∧ {𝑥 ∣ (∃𝑚 ∈ ω (dom ∅ = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(∅‘𝑚))) ∨ (dom ∅ = ∅ ∧ 𝑥 ∈ 𝐴))} ∈ V) → ((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐴))})‘∅) = {𝑥 ∣ (∃𝑚 ∈ ω (dom ∅ = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(∅‘𝑚))) ∨ (dom ∅ = ∅ ∧ 𝑥 ∈ 𝐴))})
34	20, 33	mpan 424	. . . . . 6 ⊢ ({𝑥 ∣ (∃𝑚 ∈ ω (dom ∅ = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(∅‘𝑚))) ∨ (dom ∅ = ∅ ∧ 𝑥 ∈ 𝐴))} ∈ V → ((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐴))})‘∅) = {𝑥 ∣ (∃𝑚 ∈ ω (dom ∅ = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(∅‘𝑚))) ∨ (dom ∅ = ∅ ∧ 𝑥 ∈ 𝐴))})
35	34, 17	eqtrdi 2281	. . . . 5 ⊢ ({𝑥 ∣ (∃𝑚 ∈ ω (dom ∅ = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(∅‘𝑚))) ∨ (dom ∅ = ∅ ∧ 𝑥 ∈ 𝐴))} ∈ V → ((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐴))})‘∅) = 𝐴)
36	19, 35	syl 14	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐴))})‘∅) = 𝐴)
37	36, 18	eqeltrd 2309	. . 3 ⊢ (𝐴 ∈ 𝑉 → ((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐴))})‘∅) ∈ V)
38		df-frec 6622	. . . . . 6 ⊢ frec(𝐹, 𝐴) = (recs((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐴))})) ↾ ω)
39	38	fveq1i 5671	. . . . 5 ⊢ (frec(𝐹, 𝐴)‘∅) = ((recs((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐴))})) ↾ ω)‘∅)
40		peano1 4716	. . . . . 6 ⊢ ∅ ∈ ω
41		fvres 5694	. . . . . 6 ⊢ (∅ ∈ ω → ((recs((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐴))})) ↾ ω)‘∅) = (recs((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐴))}))‘∅))
42	40, 41	ax-mp 5	. . . . 5 ⊢ ((recs((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐴))})) ↾ ω)‘∅) = (recs((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐴))}))‘∅)
43	39, 42	eqtri 2253	. . . 4 ⊢ (frec(𝐹, 𝐴)‘∅) = (recs((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐴))}))‘∅)
44		eqid 2232	. . . . 5 ⊢ recs((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐴))})) = recs((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐴))}))
45	44	tfr0 6554	. . . 4 ⊢ (((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐴))})‘∅) ∈ V → (recs((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐴))}))‘∅) = ((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐴))})‘∅))
46	43, 45	eqtrid 2277	. . 3 ⊢ (((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐴))})‘∅) ∈ V → (frec(𝐹, 𝐴)‘∅) = ((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐴))})‘∅))
47	37, 46	syl 14	. 2 ⊢ (𝐴 ∈ 𝑉 → (frec(𝐹, 𝐴)‘∅) = ((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐴))})‘∅))
48	47, 36	eqtrd 2265	1 ⊢ (𝐴 ∈ 𝑉 → (frec(𝐹, 𝐴)‘∅) = 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ∨ wo 716   = wceq 1398   ∈ wcel 2203  {cab 2218   ≠ wne 2412  ∃wrex 2521  Vcvv 2813  ∅c0 3508   ↦ cmpt 4171  suc csuc 4486  ωcom 4712  dom cdm 4749   ↾ cres 4751  ‘cfv 5352  recscrecs 6535  freccfrec 6621

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-id 4414  df-iord 4487  df-on 4489  df-suc 4492  df-iom 4713  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-res 4761  df-iota 5312  df-fun 5354  df-fn 5355  df-fv 5360  df-recs 6536  df-frec 6622

This theorem is referenced by:  frecrdg  6639  frec2uz0d  10761  frec2uzrdg  10771  frecuzrdg0  10775  frecuzrdgg  10778  frecuzrdg0t  10784  seq3val  10822  seqvalcd  10823