Theorem frec0g 6302

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 4761	. . . . . . . . . 10 ⊢ dom ∅ = ∅
2	1	biantrur 301	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐴 ↔ (dom ∅ = ∅ ∧ 𝑥 ∈ 𝐴))
3		vex 2692	. . . . . . . . . . . . . . . 16 ⊢ 𝑚 ∈ V
4		nsuceq0g 4348	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 ∈ V → suc 𝑚 ≠ ∅)
5	3, 4	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ suc 𝑚 ≠ ∅
6	5	nesymi 2355	. . . . . . . . . . . . . 14 ⊢ ¬ ∅ = suc 𝑚
7	1	eqeq1i 2148	. . . . . . . . . . . . . 14 ⊢ (dom ∅ = suc 𝑚 ↔ ∅ = suc 𝑚)
8	6, 7	mtbir 661	. . . . . . . . . . . . 13 ⊢ ¬ dom ∅ = suc 𝑚
9	8	intnanr 916	. . . . . . . . . . . 12 ⊢ ¬ (dom ∅ = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(∅‘𝑚)))
10	9	a1i 9	. . . . . . . . . . 11 ⊢ (𝑚 ∈ ω → ¬ (dom ∅ = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(∅‘𝑚))))
11	10	nrex 2527	. . . . . . . . . 10 ⊢ ¬ ∃𝑚 ∈ ω (dom ∅ = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(∅‘𝑚)))
12	11	biorfi 736	. . . . . . . . 9 ⊢ ((dom ∅ = ∅ ∧ 𝑥 ∈ 𝐴) ↔ ((dom ∅ = ∅ ∧ 𝑥 ∈ 𝐴) ∨ ∃𝑚 ∈ ω (dom ∅ = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(∅‘𝑚)))))
13		orcom 718	. . . . . . . . 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 2256	. . . . . . 7 ⊢ {𝑥 ∣ 𝑥 ∈ 𝐴} = {𝑥 ∣ (∃𝑚 ∈ ω (dom ∅ = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(∅‘𝑚))) ∨ (dom ∅ = ∅ ∧ 𝑥 ∈ 𝐴))}
16		abid2 2261	. . . . . . 7 ⊢ {𝑥 ∣ 𝑥 ∈ 𝐴} = 𝐴
17	15, 16	eqtr3i 2163	. . . . . 6 ⊢ {𝑥 ∣ (∃𝑚 ∈ ω (dom ∅ = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(∅‘𝑚))) ∨ (dom ∅ = ∅ ∧ 𝑥 ∈ 𝐴))} = 𝐴
18		elex 2700	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
19	17, 18	eqeltrid 2227	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∣ (∃𝑚 ∈ ω (dom ∅ = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(∅‘𝑚))) ∨ (dom ∅ = ∅ ∧ 𝑥 ∈ 𝐴))} ∈ V)
20		0ex 4063	. . . . . . 7 ⊢ ∅ ∈ V
21		dmeq 4747	. . . . . . . . . . . . 13 ⊢ (𝑔 = ∅ → dom 𝑔 = dom ∅)
22	21	eqeq1d 2149	. . . . . . . . . . . 12 ⊢ (𝑔 = ∅ → (dom 𝑔 = suc 𝑚 ↔ dom ∅ = suc 𝑚))
23		fveq1 5428	. . . . . . . . . . . . . 14 ⊢ (𝑔 = ∅ → (𝑔‘𝑚) = (∅‘𝑚))
24	23	fveq2d 5433	. . . . . . . . . . . . 13 ⊢ (𝑔 = ∅ → (𝐹‘(𝑔‘𝑚)) = (𝐹‘(∅‘𝑚)))
25	24	eleq2d 2210	. . . . . . . . . . . 12 ⊢ (𝑔 = ∅ → (𝑥 ∈ (𝐹‘(𝑔‘𝑚)) ↔ 𝑥 ∈ (𝐹‘(∅‘𝑚))))
26	22, 25	anbi12d 465	. . . . . . . . . . 11 ⊢ (𝑔 = ∅ → ((dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ↔ (dom ∅ = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(∅‘𝑚)))))
27	26	rexbidv 2439	. . . . . . . . . 10 ⊢ (𝑔 = ∅ → (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ↔ ∃𝑚 ∈ ω (dom ∅ = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(∅‘𝑚)))))
28	21	eqeq1d 2149	. . . . . . . . . . 11 ⊢ (𝑔 = ∅ → (dom 𝑔 = ∅ ↔ dom ∅ = ∅))
29	28	anbi1d 461	. . . . . . . . . 10 ⊢ (𝑔 = ∅ → ((dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐴) ↔ (dom ∅ = ∅ ∧ 𝑥 ∈ 𝐴)))
30	27, 29	orbi12d 783	. . . . . . . . 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 2258	. . . . . . . 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 2140	. . . . . . . 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 5505	. . . . . . 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 421	. . . . . 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 2189	. . . . 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 2217	. . 3 ⊢ (𝐴 ∈ 𝑉 → ((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐴))})‘∅) ∈ V)
38		df-frec 6296	. . . . . 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 5430	. . . . 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 4516	. . . . . 6 ⊢ ∅ ∈ ω
41		fvres 5453	. . . . . 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 2161	. . . 4 ⊢ (frec(𝐹, 𝐴)‘∅) = (recs((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐴))}))‘∅)
44		eqid 2140	. . . . 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 6228	. . . 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	syl5eq 2185	. . 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 2173	1 ⊢ (𝐴 ∈ 𝑉 → (frec(𝐹, 𝐴)‘∅) = 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ∨ wo 698   = wceq 1332   ∈ wcel 1481  {cab 2126   ≠ wne 2309  ∃wrex 2418  Vcvv 2689  ∅c0 3368   ↦ cmpt 3997  suc csuc 4295  ωcom 4512  dom cdm 4547   ↾ cres 4549  ‘cfv 5131  recscrecs 6209  freccfrec 6295

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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-nul 4062  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-int 3780  df-iun 3823  df-br 3938  df-opab 3998  df-mpt 3999  df-tr 4035  df-id 4223  df-iord 4296  df-on 4298  df-suc 4301  df-iom 4513  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-res 4559  df-iota 5096  df-fun 5133  df-fn 5134  df-fv 5139  df-recs 6210  df-frec 6296

This theorem is referenced by:  frecrdg  6313  frec2uz0d  10203  frec2uzrdg  10213  frecuzrdg0  10217  frecuzrdgg  10220  frecuzrdg0t  10226  seq3val  10262  seqvalcd  10263