Theorem frec0g 6398

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 4842	. . . . . . . . . 10 ⊢ dom ∅ = ∅
2	1	biantrur 303	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐴 ↔ (dom ∅ = ∅ ∧ 𝑥 ∈ 𝐴))
3		vex 2741	. . . . . . . . . . . . . . . 16 ⊢ 𝑚 ∈ V
4		nsuceq0g 4419	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 ∈ V → suc 𝑚 ≠ ∅)
5	3, 4	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ suc 𝑚 ≠ ∅
6	5	nesymi 2393	. . . . . . . . . . . . . 14 ⊢ ¬ ∅ = suc 𝑚
7	1	eqeq1i 2185	. . . . . . . . . . . . . 14 ⊢ (dom ∅ = suc 𝑚 ↔ ∅ = suc 𝑚)
8	6, 7	mtbir 671	. . . . . . . . . . . . 13 ⊢ ¬ dom ∅ = suc 𝑚
9	8	intnanr 930	. . . . . . . . . . . 12 ⊢ ¬ (dom ∅ = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(∅‘𝑚)))
10	9	a1i 9	. . . . . . . . . . 11 ⊢ (𝑚 ∈ ω → ¬ (dom ∅ = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(∅‘𝑚))))
11	10	nrex 2569	. . . . . . . . . 10 ⊢ ¬ ∃𝑚 ∈ ω (dom ∅ = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(∅‘𝑚)))
12	11	biorfi 746	. . . . . . . . 9 ⊢ ((dom ∅ = ∅ ∧ 𝑥 ∈ 𝐴) ↔ ((dom ∅ = ∅ ∧ 𝑥 ∈ 𝐴) ∨ ∃𝑚 ∈ ω (dom ∅ = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(∅‘𝑚)))))
13		orcom 728	. . . . . . . . 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 2293	. . . . . . 7 ⊢ {𝑥 ∣ 𝑥 ∈ 𝐴} = {𝑥 ∣ (∃𝑚 ∈ ω (dom ∅ = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(∅‘𝑚))) ∨ (dom ∅ = ∅ ∧ 𝑥 ∈ 𝐴))}
16		abid2 2298	. . . . . . 7 ⊢ {𝑥 ∣ 𝑥 ∈ 𝐴} = 𝐴
17	15, 16	eqtr3i 2200	. . . . . 6 ⊢ {𝑥 ∣ (∃𝑚 ∈ ω (dom ∅ = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(∅‘𝑚))) ∨ (dom ∅ = ∅ ∧ 𝑥 ∈ 𝐴))} = 𝐴
18		elex 2749	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
19	17, 18	eqeltrid 2264	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∣ (∃𝑚 ∈ ω (dom ∅ = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(∅‘𝑚))) ∨ (dom ∅ = ∅ ∧ 𝑥 ∈ 𝐴))} ∈ V)
20		0ex 4131	. . . . . . 7 ⊢ ∅ ∈ V
21		dmeq 4828	. . . . . . . . . . . . 13 ⊢ (𝑔 = ∅ → dom 𝑔 = dom ∅)
22	21	eqeq1d 2186	. . . . . . . . . . . 12 ⊢ (𝑔 = ∅ → (dom 𝑔 = suc 𝑚 ↔ dom ∅ = suc 𝑚))
23		fveq1 5515	. . . . . . . . . . . . . 14 ⊢ (𝑔 = ∅ → (𝑔‘𝑚) = (∅‘𝑚))
24	23	fveq2d 5520	. . . . . . . . . . . . 13 ⊢ (𝑔 = ∅ → (𝐹‘(𝑔‘𝑚)) = (𝐹‘(∅‘𝑚)))
25	24	eleq2d 2247	. . . . . . . . . . . 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 2478	. . . . . . . . . 10 ⊢ (𝑔 = ∅ → (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ↔ ∃𝑚 ∈ ω (dom ∅ = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(∅‘𝑚)))))
28	21	eqeq1d 2186	. . . . . . . . . . 11 ⊢ (𝑔 = ∅ → (dom 𝑔 = ∅ ↔ dom ∅ = ∅))
29	28	anbi1d 465	. . . . . . . . . 10 ⊢ (𝑔 = ∅ → ((dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐴) ↔ (dom ∅ = ∅ ∧ 𝑥 ∈ 𝐴)))
30	27, 29	orbi12d 793	. . . . . . . . 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 2295	. . . . . . . 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 2177	. . . . . . . 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 5593	. . . . . . 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 2226	. . . . 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 2254	. . 3 ⊢ (𝐴 ∈ 𝑉 → ((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐴))})‘∅) ∈ V)
38		df-frec 6392	. . . . . 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 5517	. . . . 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 4594	. . . . . 6 ⊢ ∅ ∈ ω
41		fvres 5540	. . . . . 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 2198	. . . 4 ⊢ (frec(𝐹, 𝐴)‘∅) = (recs((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐴))}))‘∅)
44		eqid 2177	. . . . 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 6324	. . . 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 2222	. . 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 2210	1 ⊢ (𝐴 ∈ 𝑉 → (frec(𝐹, 𝐴)‘∅) = 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ∨ wo 708   = wceq 1353   ∈ wcel 2148  {cab 2163   ≠ wne 2347  ∃wrex 2456  Vcvv 2738  ∅c0 3423   ↦ cmpt 4065  suc csuc 4366  ωcom 4590  dom cdm 4627   ↾ cres 4629  ‘cfv 5217  recscrecs 6305  freccfrec 6391

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-nul 4130  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-tr 4103  df-id 4294  df-iord 4367  df-on 4369  df-suc 4372  df-iom 4591  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-res 4639  df-iota 5179  df-fun 5219  df-fn 5220  df-fv 5225  df-recs 6306  df-frec 6392

This theorem is referenced by:  frecrdg  6409  frec2uz0d  10399  frec2uzrdg  10409  frecuzrdg0  10413  frecuzrdgg  10416  frecuzrdg0t  10422  seq3val  10458  seqvalcd  10459