Theorem frec0g 6450

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 4876	. . . . . . . . . 10 ⊢ dom ∅ = ∅
2	1	biantrur 303	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐴 ↔ (dom ∅ = ∅ ∧ 𝑥 ∈ 𝐴))
3		vex 2763	. . . . . . . . . . . . . . . 16 ⊢ 𝑚 ∈ V
4		nsuceq0g 4449	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 ∈ V → suc 𝑚 ≠ ∅)
5	3, 4	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ suc 𝑚 ≠ ∅
6	5	nesymi 2410	. . . . . . . . . . . . . 14 ⊢ ¬ ∅ = suc 𝑚
7	1	eqeq1i 2201	. . . . . . . . . . . . . 14 ⊢ (dom ∅ = suc 𝑚 ↔ ∅ = suc 𝑚)
8	6, 7	mtbir 672	. . . . . . . . . . . . 13 ⊢ ¬ dom ∅ = suc 𝑚
9	8	intnanr 931	. . . . . . . . . . . 12 ⊢ ¬ (dom ∅ = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(∅‘𝑚)))
10	9	a1i 9	. . . . . . . . . . 11 ⊢ (𝑚 ∈ ω → ¬ (dom ∅ = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(∅‘𝑚))))
11	10	nrex 2586	. . . . . . . . . 10 ⊢ ¬ ∃𝑚 ∈ ω (dom ∅ = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(∅‘𝑚)))
12	11	biorfi 747	. . . . . . . . 9 ⊢ ((dom ∅ = ∅ ∧ 𝑥 ∈ 𝐴) ↔ ((dom ∅ = ∅ ∧ 𝑥 ∈ 𝐴) ∨ ∃𝑚 ∈ ω (dom ∅ = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(∅‘𝑚)))))
13		orcom 729	. . . . . . . . 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 2309	. . . . . . 7 ⊢ {𝑥 ∣ 𝑥 ∈ 𝐴} = {𝑥 ∣ (∃𝑚 ∈ ω (dom ∅ = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(∅‘𝑚))) ∨ (dom ∅ = ∅ ∧ 𝑥 ∈ 𝐴))}
16		abid2 2314	. . . . . . 7 ⊢ {𝑥 ∣ 𝑥 ∈ 𝐴} = 𝐴
17	15, 16	eqtr3i 2216	. . . . . 6 ⊢ {𝑥 ∣ (∃𝑚 ∈ ω (dom ∅ = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(∅‘𝑚))) ∨ (dom ∅ = ∅ ∧ 𝑥 ∈ 𝐴))} = 𝐴
18		elex 2771	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
19	17, 18	eqeltrid 2280	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∣ (∃𝑚 ∈ ω (dom ∅ = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(∅‘𝑚))) ∨ (dom ∅ = ∅ ∧ 𝑥 ∈ 𝐴))} ∈ V)
20		0ex 4156	. . . . . . 7 ⊢ ∅ ∈ V
21		dmeq 4862	. . . . . . . . . . . . 13 ⊢ (𝑔 = ∅ → dom 𝑔 = dom ∅)
22	21	eqeq1d 2202	. . . . . . . . . . . 12 ⊢ (𝑔 = ∅ → (dom 𝑔 = suc 𝑚 ↔ dom ∅ = suc 𝑚))
23		fveq1 5553	. . . . . . . . . . . . . 14 ⊢ (𝑔 = ∅ → (𝑔‘𝑚) = (∅‘𝑚))
24	23	fveq2d 5558	. . . . . . . . . . . . 13 ⊢ (𝑔 = ∅ → (𝐹‘(𝑔‘𝑚)) = (𝐹‘(∅‘𝑚)))
25	24	eleq2d 2263	. . . . . . . . . . . 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 2495	. . . . . . . . . 10 ⊢ (𝑔 = ∅ → (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ↔ ∃𝑚 ∈ ω (dom ∅ = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(∅‘𝑚)))))
28	21	eqeq1d 2202	. . . . . . . . . . 11 ⊢ (𝑔 = ∅ → (dom 𝑔 = ∅ ↔ dom ∅ = ∅))
29	28	anbi1d 465	. . . . . . . . . 10 ⊢ (𝑔 = ∅ → ((dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐴) ↔ (dom ∅ = ∅ ∧ 𝑥 ∈ 𝐴)))
30	27, 29	orbi12d 794	. . . . . . . . 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 2311	. . . . . . . 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 2193	. . . . . . . 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 5633	. . . . . . 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 2242	. . . . 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 2270	. . 3 ⊢ (𝐴 ∈ 𝑉 → ((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐴))})‘∅) ∈ V)
38		df-frec 6444	. . . . . 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 5555	. . . . 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 4626	. . . . . 6 ⊢ ∅ ∈ ω
41		fvres 5578	. . . . . 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 2214	. . . 4 ⊢ (frec(𝐹, 𝐴)‘∅) = (recs((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐴))}))‘∅)
44		eqid 2193	. . . . 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 6376	. . . 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 2238	. . 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 2226	1 ⊢ (𝐴 ∈ 𝑉 → (frec(𝐹, 𝐴)‘∅) = 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ∨ wo 709   = wceq 1364   ∈ wcel 2164  {cab 2179   ≠ wne 2364  ∃wrex 2473  Vcvv 2760  ∅c0 3446   ↦ cmpt 4090  suc csuc 4396  ωcom 4622  dom cdm 4659   ↾ cres 4661  ‘cfv 5254  recscrecs 6357  freccfrec 6443

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4324  df-iord 4397  df-on 4399  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-res 4671  df-iota 5215  df-fun 5256  df-fn 5257  df-fv 5262  df-recs 6358  df-frec 6444

This theorem is referenced by:  frecrdg  6461  frec2uz0d  10470  frec2uzrdg  10480  frecuzrdg0  10484  frecuzrdgg  10487  frecuzrdg0t  10493  seq3val  10531  seqvalcd  10532