Theorem frec0g 6376

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 4825	. . . . . . . . . 10 ⊢ dom ∅ = ∅
2	1	biantrur 301	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐴 ↔ (dom ∅ = ∅ ∧ 𝑥 ∈ 𝐴))
3		vex 2733	. . . . . . . . . . . . . . . 16 ⊢ 𝑚 ∈ V
4		nsuceq0g 4403	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 ∈ V → suc 𝑚 ≠ ∅)
5	3, 4	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ suc 𝑚 ≠ ∅
6	5	nesymi 2386	. . . . . . . . . . . . . 14 ⊢ ¬ ∅ = suc 𝑚
7	1	eqeq1i 2178	. . . . . . . . . . . . . 14 ⊢ (dom ∅ = suc 𝑚 ↔ ∅ = suc 𝑚)
8	6, 7	mtbir 666	. . . . . . . . . . . . 13 ⊢ ¬ dom ∅ = suc 𝑚
9	8	intnanr 925	. . . . . . . . . . . 12 ⊢ ¬ (dom ∅ = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(∅‘𝑚)))
10	9	a1i 9	. . . . . . . . . . 11 ⊢ (𝑚 ∈ ω → ¬ (dom ∅ = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(∅‘𝑚))))
11	10	nrex 2562	. . . . . . . . . 10 ⊢ ¬ ∃𝑚 ∈ ω (dom ∅ = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(∅‘𝑚)))
12	11	biorfi 741	. . . . . . . . 9 ⊢ ((dom ∅ = ∅ ∧ 𝑥 ∈ 𝐴) ↔ ((dom ∅ = ∅ ∧ 𝑥 ∈ 𝐴) ∨ ∃𝑚 ∈ ω (dom ∅ = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(∅‘𝑚)))))
13		orcom 723	. . . . . . . . 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 205	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐴 ↔ (∃𝑚 ∈ ω (dom ∅ = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(∅‘𝑚))) ∨ (dom ∅ = ∅ ∧ 𝑥 ∈ 𝐴)))
15	14	abbii 2286	. . . . . . 7 ⊢ {𝑥 ∣ 𝑥 ∈ 𝐴} = {𝑥 ∣ (∃𝑚 ∈ ω (dom ∅ = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(∅‘𝑚))) ∨ (dom ∅ = ∅ ∧ 𝑥 ∈ 𝐴))}
16		abid2 2291	. . . . . . 7 ⊢ {𝑥 ∣ 𝑥 ∈ 𝐴} = 𝐴
17	15, 16	eqtr3i 2193	. . . . . 6 ⊢ {𝑥 ∣ (∃𝑚 ∈ ω (dom ∅ = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(∅‘𝑚))) ∨ (dom ∅ = ∅ ∧ 𝑥 ∈ 𝐴))} = 𝐴
18		elex 2741	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
19	17, 18	eqeltrid 2257	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∣ (∃𝑚 ∈ ω (dom ∅ = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(∅‘𝑚))) ∨ (dom ∅ = ∅ ∧ 𝑥 ∈ 𝐴))} ∈ V)
20		0ex 4116	. . . . . . 7 ⊢ ∅ ∈ V
21		dmeq 4811	. . . . . . . . . . . . 13 ⊢ (𝑔 = ∅ → dom 𝑔 = dom ∅)
22	21	eqeq1d 2179	. . . . . . . . . . . 12 ⊢ (𝑔 = ∅ → (dom 𝑔 = suc 𝑚 ↔ dom ∅ = suc 𝑚))
23		fveq1 5495	. . . . . . . . . . . . . 14 ⊢ (𝑔 = ∅ → (𝑔‘𝑚) = (∅‘𝑚))
24	23	fveq2d 5500	. . . . . . . . . . . . 13 ⊢ (𝑔 = ∅ → (𝐹‘(𝑔‘𝑚)) = (𝐹‘(∅‘𝑚)))
25	24	eleq2d 2240	. . . . . . . . . . . 12 ⊢ (𝑔 = ∅ → (𝑥 ∈ (𝐹‘(𝑔‘𝑚)) ↔ 𝑥 ∈ (𝐹‘(∅‘𝑚))))
26	22, 25	anbi12d 470	. . . . . . . . . . 11 ⊢ (𝑔 = ∅ → ((dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ↔ (dom ∅ = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(∅‘𝑚)))))
27	26	rexbidv 2471	. . . . . . . . . 10 ⊢ (𝑔 = ∅ → (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ↔ ∃𝑚 ∈ ω (dom ∅ = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(∅‘𝑚)))))
28	21	eqeq1d 2179	. . . . . . . . . . 11 ⊢ (𝑔 = ∅ → (dom 𝑔 = ∅ ↔ dom ∅ = ∅))
29	28	anbi1d 462	. . . . . . . . . 10 ⊢ (𝑔 = ∅ → ((dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐴) ↔ (dom ∅ = ∅ ∧ 𝑥 ∈ 𝐴)))
30	27, 29	orbi12d 788	. . . . . . . . 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 2288	. . . . . . . 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 2170	. . . . . . . 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 5572	. . . . . . 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 422	. . . . . 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 2219	. . . . 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 2247	. . 3 ⊢ (𝐴 ∈ 𝑉 → ((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐴))})‘∅) ∈ V)
38		df-frec 6370	. . . . . 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 5497	. . . . 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 4578	. . . . . 6 ⊢ ∅ ∈ ω
41		fvres 5520	. . . . . 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 2191	. . . 4 ⊢ (frec(𝐹, 𝐴)‘∅) = (recs((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐴))}))‘∅)
44		eqid 2170	. . . . 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 6302	. . . 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 2215	. . 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 2203	1 ⊢ (𝐴 ∈ 𝑉 → (frec(𝐹, 𝐴)‘∅) = 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ∨ wo 703   = wceq 1348   ∈ wcel 2141  {cab 2156   ≠ wne 2340  ∃wrex 2449  Vcvv 2730  ∅c0 3414   ↦ cmpt 4050  suc csuc 4350  ωcom 4574  dom cdm 4611   ↾ cres 4613  ‘cfv 5198  recscrecs 6283  freccfrec 6369

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-iord 4351  df-on 4353  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-res 4623  df-iota 5160  df-fun 5200  df-fn 5201  df-fv 5206  df-recs 6284  df-frec 6370

This theorem is referenced by:  frecrdg  6387  frec2uz0d  10355  frec2uzrdg  10365  frecuzrdg0  10369  frecuzrdgg  10372  frecuzrdg0t  10378  seq3val  10414  seqvalcd  10415