Theorem freccl 5993

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 5178	. . . . 5 ⊢ (𝑥 = 𝐵 → (frec(𝐹, 𝐴)‘𝑥) = (frec(𝐹, 𝐴)‘𝐵))
3	2	eleq1d 2106	. . . 4 ⊢ (𝑥 = 𝐵 → ((frec(𝐹, 𝐴)‘𝑥) ∈ 𝑆 ↔ (frec(𝐹, 𝐴)‘𝐵) ∈ 𝑆))
4	3	imbi2d 219	. . 3 ⊢ (𝑥 = 𝐵 → ((𝜑 → (frec(𝐹, 𝐴)‘𝑥) ∈ 𝑆) ↔ (𝜑 → (frec(𝐹, 𝐴)‘𝐵) ∈ 𝑆)))
5		fveq2 5178	. . . . 5 ⊢ (𝑥 = ∅ → (frec(𝐹, 𝐴)‘𝑥) = (frec(𝐹, 𝐴)‘∅))
6	5	eleq1d 2106	. . . 4 ⊢ (𝑥 = ∅ → ((frec(𝐹, 𝐴)‘𝑥) ∈ 𝑆 ↔ (frec(𝐹, 𝐴)‘∅) ∈ 𝑆))
7		fveq2 5178	. . . . 5 ⊢ (𝑥 = 𝑦 → (frec(𝐹, 𝐴)‘𝑥) = (frec(𝐹, 𝐴)‘𝑦))
8	7	eleq1d 2106	. . . 4 ⊢ (𝑥 = 𝑦 → ((frec(𝐹, 𝐴)‘𝑥) ∈ 𝑆 ↔ (frec(𝐹, 𝐴)‘𝑦) ∈ 𝑆))
9		fveq2 5178	. . . . 5 ⊢ (𝑥 = suc 𝑦 → (frec(𝐹, 𝐴)‘𝑥) = (frec(𝐹, 𝐴)‘suc 𝑦))
10	9	eleq1d 2106	. . . 4 ⊢ (𝑥 = suc 𝑦 → ((frec(𝐹, 𝐴)‘𝑥) ∈ 𝑆 ↔ (frec(𝐹, 𝐴)‘suc 𝑦) ∈ 𝑆))
11		freccl.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝑆)
12		frec0g 5983	. . . . . 6 ⊢ (𝐴 ∈ 𝑆 → (frec(𝐹, 𝐴)‘∅) = 𝐴)
13	11, 12	syl 14	. . . . 5 ⊢ (𝜑 → (frec(𝐹, 𝐴)‘∅) = 𝐴)
14	13, 11	eqeltrd 2114	. . . 4 ⊢ (𝜑 → (frec(𝐹, 𝐴)‘∅) ∈ 𝑆)
15		freccl.ex	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑧(𝐹‘𝑧) ∈ V)
16		frecfnom 5986	. . . . . . . . . 10 ⊢ ((∀𝑧(𝐹‘𝑧) ∈ V ∧ 𝐴 ∈ 𝑆) → frec(𝐹, 𝐴) Fn ω)
17	15, 11, 16	syl2anc 391	. . . . . . . . 9 ⊢ (𝜑 → frec(𝐹, 𝐴) Fn ω)
18		funfvex 5192	. . . . . . . . . 10 ⊢ ((Fun frec(𝐹, 𝐴) ∧ 𝑦 ∈ dom frec(𝐹, 𝐴)) → (frec(𝐹, 𝐴)‘𝑦) ∈ V)
19	18	funfni 4999	. . . . . . . . 9 ⊢ ((frec(𝐹, 𝐴) Fn ω ∧ 𝑦 ∈ ω) → (frec(𝐹, 𝐴)‘𝑦) ∈ V)
20	17, 19	sylan 267	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ω) → (frec(𝐹, 𝐴)‘𝑦) ∈ V)
21		isset 2561	. . . . . . . 8 ⊢ ((frec(𝐹, 𝐴)‘𝑦) ∈ V ↔ ∃𝑧 𝑧 = (frec(𝐹, 𝐴)‘𝑦))
22	20, 21	sylib 127	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ω) → ∃𝑧 𝑧 = (frec(𝐹, 𝐴)‘𝑦))
23		freccl.cl	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → (𝐹‘𝑧) ∈ 𝑆)
24	23	ex 108	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑧 ∈ 𝑆 → (𝐹‘𝑧) ∈ 𝑆))
25	24	adantr 261	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 = (frec(𝐹, 𝐴)‘𝑦)) → (𝑧 ∈ 𝑆 → (𝐹‘𝑧) ∈ 𝑆))
26		eleq1 2100	. . . . . . . . . . . 12 ⊢ (𝑧 = (frec(𝐹, 𝐴)‘𝑦) → (𝑧 ∈ 𝑆 ↔ (frec(𝐹, 𝐴)‘𝑦) ∈ 𝑆))
27	26	adantl 262	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 = (frec(𝐹, 𝐴)‘𝑦)) → (𝑧 ∈ 𝑆 ↔ (frec(𝐹, 𝐴)‘𝑦) ∈ 𝑆))
28		fveq2 5178	. . . . . . . . . . . . 13 ⊢ (𝑧 = (frec(𝐹, 𝐴)‘𝑦) → (𝐹‘𝑧) = (𝐹‘(frec(𝐹, 𝐴)‘𝑦)))
29	28	eleq1d 2106	. . . . . . . . . . . 12 ⊢ (𝑧 = (frec(𝐹, 𝐴)‘𝑦) → ((𝐹‘𝑧) ∈ 𝑆 ↔ (𝐹‘(frec(𝐹, 𝐴)‘𝑦)) ∈ 𝑆))
30	29	adantl 262	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 = (frec(𝐹, 𝐴)‘𝑦)) → ((𝐹‘𝑧) ∈ 𝑆 ↔ (𝐹‘(frec(𝐹, 𝐴)‘𝑦)) ∈ 𝑆))
31	25, 27, 30	3imtr3d 191	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 = (frec(𝐹, 𝐴)‘𝑦)) → ((frec(𝐹, 𝐴)‘𝑦) ∈ 𝑆 → (𝐹‘(frec(𝐹, 𝐴)‘𝑦)) ∈ 𝑆))
32	31	ex 108	. . . . . . . . 9 ⊢ (𝜑 → (𝑧 = (frec(𝐹, 𝐴)‘𝑦) → ((frec(𝐹, 𝐴)‘𝑦) ∈ 𝑆 → (𝐹‘(frec(𝐹, 𝐴)‘𝑦)) ∈ 𝑆)))
33	32	exlimdv 1700	. . . . . . . 8 ⊢ (𝜑 → (∃𝑧 𝑧 = (frec(𝐹, 𝐴)‘𝑦) → ((frec(𝐹, 𝐴)‘𝑦) ∈ 𝑆 → (𝐹‘(frec(𝐹, 𝐴)‘𝑦)) ∈ 𝑆)))
34	33	adantr 261	. . . . . . 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 261	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ω) → ∀𝑧(𝐹‘𝑧) ∈ V)
37	11	adantr 261	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ω) → 𝐴 ∈ 𝑆)
38		simpr 103	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ω) → 𝑦 ∈ ω)
39		frecsuc 5991	. . . . . . . 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 1135	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ω) → (frec(𝐹, 𝐴)‘suc 𝑦) = (𝐹‘(frec(𝐹, 𝐴)‘𝑦)))
41	40	eleq1d 2106	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ω) → ((frec(𝐹, 𝐴)‘suc 𝑦) ∈ 𝑆 ↔ (𝐹‘(frec(𝐹, 𝐴)‘𝑦)) ∈ 𝑆))
42	35, 41	sylibrd 158	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ω) → ((frec(𝐹, 𝐴)‘𝑦) ∈ 𝑆 → (frec(𝐹, 𝐴)‘suc 𝑦) ∈ 𝑆))
43	42	expcom 109	. . . 4 ⊢ (𝑦 ∈ ω → (𝜑 → ((frec(𝐹, 𝐴)‘𝑦) ∈ 𝑆 → (frec(𝐹, 𝐴)‘suc 𝑦) ∈ 𝑆)))
44	6, 8, 10, 14, 43	finds2 4324	. . 3 ⊢ (𝑥 ∈ ω → (𝜑 → (frec(𝐹, 𝐴)‘𝑥) ∈ 𝑆))
45	4, 44	vtoclga 2619	. 2 ⊢ (𝐵 ∈ ω → (𝜑 → (frec(𝐹, 𝐴)‘𝐵) ∈ 𝑆))
46	1, 45	mpcom 32	1 ⊢ (𝜑 → (frec(𝐹, 𝐴)‘𝐵) ∈ 𝑆)

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98  ∀wal 1241   = wceq 1243  ∃wex 1381   ∈ wcel 1393  Vcvv 2557  ∅c0 3224  suc csuc 4102  ωcom 4313   Fn wfn 4897  ‘cfv 4902  freccfrec 5977

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-id 4030  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-recs 5920  df-frec 5978

This theorem is referenced by:  frecuzrdgrrn  9194