Theorem freccl 6016

Description: Closure for finite recursion. (Contributed by Jim Kingdon, 25-May-2020.)

Hypotheses

Ref	Expression
freccl.ex	⊢ (𝜑 → ∀𝑧(𝐹‘𝑧) ∈ V)
freccl.a	⊢ (𝜑 → 𝐴 ∈ 𝑆)
freccl.cl	⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → (𝐹‘𝑧) ∈ 𝑆)
freccl.b	⊢ (𝜑 → 𝐵 ∈ ω)

Assertion

Ref	Expression
freccl	⊢ (𝜑 → (frec(𝐹, 𝐴)‘𝐵) ∈ 𝑆)

Distinct variable groups:   𝑧,𝐴   𝑧,𝐹   𝑧,𝑆   𝜑,𝑧

Allowed substitution hint:   𝐵(𝑧)

Proof of Theorem freccl

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		freccl.b	. 2 ⊢ (𝜑 → 𝐵 ∈ ω)
2		fveq2 5198	. . . . 5 ⊢ (𝑥 = 𝐵 → (frec(𝐹, 𝐴)‘𝑥) = (frec(𝐹, 𝐴)‘𝐵))
3	2	eleq1d 2147	. . . 4 ⊢ (𝑥 = 𝐵 → ((frec(𝐹, 𝐴)‘𝑥) ∈ 𝑆 ↔ (frec(𝐹, 𝐴)‘𝐵) ∈ 𝑆))
4	3	imbi2d 228	. . 3 ⊢ (𝑥 = 𝐵 → ((𝜑 → (frec(𝐹, 𝐴)‘𝑥) ∈ 𝑆) ↔ (𝜑 → (frec(𝐹, 𝐴)‘𝐵) ∈ 𝑆)))
5		fveq2 5198	. . . . 5 ⊢ (𝑥 = ∅ → (frec(𝐹, 𝐴)‘𝑥) = (frec(𝐹, 𝐴)‘∅))
6	5	eleq1d 2147	. . . 4 ⊢ (𝑥 = ∅ → ((frec(𝐹, 𝐴)‘𝑥) ∈ 𝑆 ↔ (frec(𝐹, 𝐴)‘∅) ∈ 𝑆))
7		fveq2 5198	. . . . 5 ⊢ (𝑥 = 𝑦 → (frec(𝐹, 𝐴)‘𝑥) = (frec(𝐹, 𝐴)‘𝑦))
8	7	eleq1d 2147	. . . 4 ⊢ (𝑥 = 𝑦 → ((frec(𝐹, 𝐴)‘𝑥) ∈ 𝑆 ↔ (frec(𝐹, 𝐴)‘𝑦) ∈ 𝑆))
9		fveq2 5198	. . . . 5 ⊢ (𝑥 = suc 𝑦 → (frec(𝐹, 𝐴)‘𝑥) = (frec(𝐹, 𝐴)‘suc 𝑦))
10	9	eleq1d 2147	. . . 4 ⊢ (𝑥 = suc 𝑦 → ((frec(𝐹, 𝐴)‘𝑥) ∈ 𝑆 ↔ (frec(𝐹, 𝐴)‘suc 𝑦) ∈ 𝑆))
11		freccl.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝑆)
12		frec0g 6006	. . . . . 6 ⊢ (𝐴 ∈ 𝑆 → (frec(𝐹, 𝐴)‘∅) = 𝐴)
13	11, 12	syl 14	. . . . 5 ⊢ (𝜑 → (frec(𝐹, 𝐴)‘∅) = 𝐴)
14	13, 11	eqeltrd 2155	. . . 4 ⊢ (𝜑 → (frec(𝐹, 𝐴)‘∅) ∈ 𝑆)
15		freccl.ex	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑧(𝐹‘𝑧) ∈ V)
16		frecfnom 6009	. . . . . . . . . 10 ⊢ ((∀𝑧(𝐹‘𝑧) ∈ V ∧ 𝐴 ∈ 𝑆) → frec(𝐹, 𝐴) Fn ω)
17	15, 11, 16	syl2anc 403	. . . . . . . . 9 ⊢ (𝜑 → frec(𝐹, 𝐴) Fn ω)
18		funfvex 5212	. . . . . . . . . 10 ⊢ ((Fun frec(𝐹, 𝐴) ∧ 𝑦 ∈ dom frec(𝐹, 𝐴)) → (frec(𝐹, 𝐴)‘𝑦) ∈ V)
19	18	funfni 5019	. . . . . . . . 9 ⊢ ((frec(𝐹, 𝐴) Fn ω ∧ 𝑦 ∈ ω) → (frec(𝐹, 𝐴)‘𝑦) ∈ V)
20	17, 19	sylan 277	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ω) → (frec(𝐹, 𝐴)‘𝑦) ∈ V)
21		isset 2605	. . . . . . . 8 ⊢ ((frec(𝐹, 𝐴)‘𝑦) ∈ V ↔ ∃𝑧 𝑧 = (frec(𝐹, 𝐴)‘𝑦))
22	20, 21	sylib 120	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ω) → ∃𝑧 𝑧 = (frec(𝐹, 𝐴)‘𝑦))
23		freccl.cl	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → (𝐹‘𝑧) ∈ 𝑆)
24	23	ex 113	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑧 ∈ 𝑆 → (𝐹‘𝑧) ∈ 𝑆))
25	24	adantr 270	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 = (frec(𝐹, 𝐴)‘𝑦)) → (𝑧 ∈ 𝑆 → (𝐹‘𝑧) ∈ 𝑆))
26		eleq1 2141	. . . . . . . . . . . 12 ⊢ (𝑧 = (frec(𝐹, 𝐴)‘𝑦) → (𝑧 ∈ 𝑆 ↔ (frec(𝐹, 𝐴)‘𝑦) ∈ 𝑆))
27	26	adantl 271	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 = (frec(𝐹, 𝐴)‘𝑦)) → (𝑧 ∈ 𝑆 ↔ (frec(𝐹, 𝐴)‘𝑦) ∈ 𝑆))
28		fveq2 5198	. . . . . . . . . . . . 13 ⊢ (𝑧 = (frec(𝐹, 𝐴)‘𝑦) → (𝐹‘𝑧) = (𝐹‘(frec(𝐹, 𝐴)‘𝑦)))
29	28	eleq1d 2147	. . . . . . . . . . . 12 ⊢ (𝑧 = (frec(𝐹, 𝐴)‘𝑦) → ((𝐹‘𝑧) ∈ 𝑆 ↔ (𝐹‘(frec(𝐹, 𝐴)‘𝑦)) ∈ 𝑆))
30	29	adantl 271	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 = (frec(𝐹, 𝐴)‘𝑦)) → ((𝐹‘𝑧) ∈ 𝑆 ↔ (𝐹‘(frec(𝐹, 𝐴)‘𝑦)) ∈ 𝑆))
31	25, 27, 30	3imtr3d 200	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 = (frec(𝐹, 𝐴)‘𝑦)) → ((frec(𝐹, 𝐴)‘𝑦) ∈ 𝑆 → (𝐹‘(frec(𝐹, 𝐴)‘𝑦)) ∈ 𝑆))
32	31	ex 113	. . . . . . . . 9 ⊢ (𝜑 → (𝑧 = (frec(𝐹, 𝐴)‘𝑦) → ((frec(𝐹, 𝐴)‘𝑦) ∈ 𝑆 → (𝐹‘(frec(𝐹, 𝐴)‘𝑦)) ∈ 𝑆)))
33	32	exlimdv 1740	. . . . . . . 8 ⊢ (𝜑 → (∃𝑧 𝑧 = (frec(𝐹, 𝐴)‘𝑦) → ((frec(𝐹, 𝐴)‘𝑦) ∈ 𝑆 → (𝐹‘(frec(𝐹, 𝐴)‘𝑦)) ∈ 𝑆)))
34	33	adantr 270	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ω) → (∃𝑧 𝑧 = (frec(𝐹, 𝐴)‘𝑦) → ((frec(𝐹, 𝐴)‘𝑦) ∈ 𝑆 → (𝐹‘(frec(𝐹, 𝐴)‘𝑦)) ∈ 𝑆)))
35	22, 34	mpd 13	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ω) → ((frec(𝐹, 𝐴)‘𝑦) ∈ 𝑆 → (𝐹‘(frec(𝐹, 𝐴)‘𝑦)) ∈ 𝑆))
36	15	adantr 270	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ω) → ∀𝑧(𝐹‘𝑧) ∈ V)
37	11	adantr 270	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ω) → 𝐴 ∈ 𝑆)
38		simpr 108	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ω) → 𝑦 ∈ ω)
39		frecsuc 6014	. . . . . . . 8 ⊢ ((∀𝑧(𝐹‘𝑧) ∈ V ∧ 𝐴 ∈ 𝑆 ∧ 𝑦 ∈ ω) → (frec(𝐹, 𝐴)‘suc 𝑦) = (𝐹‘(frec(𝐹, 𝐴)‘𝑦)))
40	36, 37, 38, 39	syl3anc 1169	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ω) → (frec(𝐹, 𝐴)‘suc 𝑦) = (𝐹‘(frec(𝐹, 𝐴)‘𝑦)))
41	40	eleq1d 2147	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ω) → ((frec(𝐹, 𝐴)‘suc 𝑦) ∈ 𝑆 ↔ (𝐹‘(frec(𝐹, 𝐴)‘𝑦)) ∈ 𝑆))
42	35, 41	sylibrd 167	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ω) → ((frec(𝐹, 𝐴)‘𝑦) ∈ 𝑆 → (frec(𝐹, 𝐴)‘suc 𝑦) ∈ 𝑆))
43	42	expcom 114	. . . 4 ⊢ (𝑦 ∈ ω → (𝜑 → ((frec(𝐹, 𝐴)‘𝑦) ∈ 𝑆 → (frec(𝐹, 𝐴)‘suc 𝑦) ∈ 𝑆)))
44	6, 8, 10, 14, 43	finds2 4342	. . 3 ⊢ (𝑥 ∈ ω → (𝜑 → (frec(𝐹, 𝐴)‘𝑥) ∈ 𝑆))
45	4, 44	vtoclga 2664	. 2 ⊢ (𝐵 ∈ ω → (𝜑 → (frec(𝐹, 𝐴)‘𝐵) ∈ 𝑆))
46	1, 45	mpcom 36	1 ⊢ (𝜑 → (frec(𝐹, 𝐴)‘𝐵) ∈ 𝑆)

Syntax hints:   → wi 4   ∧ wa 102   ↔ wb 103  ∀wal 1282   = wceq 1284  ∃wex 1421   ∈ wcel 1433  Vcvv 2601  ∅c0 3251  suc csuc 4120  ωcom 4331   Fn wfn 4917  ‘cfv 4922  freccfrec 6000

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-coll 3893  ax-sep 3896  ax-nul 3904  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-iinf 4329

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-ral 2353  df-rex 2354  df-reu 2355  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-int 3637  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-tr 3876  df-id 4048  df-iord 4121  df-on 4123  df-suc 4126  df-iom 4332  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-recs 5943  df-frec 6001

This theorem is referenced by:  frecuzrdgrrn  9410