Theorem frecrdg 6617

Description: Transfinite recursion restricted to omega.

Given a suitable characteristic function, df-frec 6600 produces the same results as df-irdg 6579 restricted to ω.

Presumably the theorem would also hold if 𝐹 Fn V were changed to ∀𝑧(𝐹‘𝑧) ∈ V. (Contributed by Jim Kingdon, 29-Aug-2019.)

Hypotheses

Ref	Expression
frecrdg.1	⊢ (𝜑 → 𝐹 Fn V)
frecrdg.2	⊢ (𝜑 → 𝐴 ∈ 𝑉)
frecrdg.inc	⊢ (𝜑 → ∀𝑥 𝑥 ⊆ (𝐹‘𝑥))

Assertion

Ref	Expression
frecrdg	⊢ (𝜑 → frec(𝐹, 𝐴) = (rec(𝐹, 𝐴) ↾ ω))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐹   𝑥,𝑉   𝜑,𝑥

Proof of Theorem frecrdg

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

Step	Hyp	Ref	Expression
1		frecrdg.1	. . . 4 ⊢ (𝜑 → 𝐹 Fn V)
2		vex 2806	. . . . . 6 ⊢ 𝑧 ∈ V
3		funfvex 5665	. . . . . . 7 ⊢ ((Fun 𝐹 ∧ 𝑧 ∈ dom 𝐹) → (𝐹‘𝑧) ∈ V)
4	3	funfni 5439	. . . . . 6 ⊢ ((𝐹 Fn V ∧ 𝑧 ∈ V) → (𝐹‘𝑧) ∈ V)
5	2, 4	mpan2 425	. . . . 5 ⊢ (𝐹 Fn V → (𝐹‘𝑧) ∈ V)
6	5	alrimiv 1922	. . . 4 ⊢ (𝐹 Fn V → ∀𝑧(𝐹‘𝑧) ∈ V)
7	1, 6	syl 14	. . 3 ⊢ (𝜑 → ∀𝑧(𝐹‘𝑧) ∈ V)
8		frecrdg.2	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
9		frecfnom 6610	. . 3 ⊢ ((∀𝑧(𝐹‘𝑧) ∈ V ∧ 𝐴 ∈ 𝑉) → frec(𝐹, 𝐴) Fn ω)
10	7, 8, 9	syl2anc 411	. 2 ⊢ (𝜑 → frec(𝐹, 𝐴) Fn ω)
11		rdgifnon2 6589	. . . 4 ⊢ ((∀𝑧(𝐹‘𝑧) ∈ V ∧ 𝐴 ∈ 𝑉) → rec(𝐹, 𝐴) Fn On)
12	7, 8, 11	syl2anc 411	. . 3 ⊢ (𝜑 → rec(𝐹, 𝐴) Fn On)
13		omsson 4717	. . 3 ⊢ ω ⊆ On
14		fnssres 5452	. . 3 ⊢ ((rec(𝐹, 𝐴) Fn On ∧ ω ⊆ On) → (rec(𝐹, 𝐴) ↾ ω) Fn ω)
15	12, 13, 14	sylancl 413	. 2 ⊢ (𝜑 → (rec(𝐹, 𝐴) ↾ ω) Fn ω)
16		fveq2 5648	. . . . 5 ⊢ (𝑥 = ∅ → (frec(𝐹, 𝐴)‘𝑥) = (frec(𝐹, 𝐴)‘∅))
17		fveq2 5648	. . . . 5 ⊢ (𝑥 = ∅ → ((rec(𝐹, 𝐴) ↾ ω)‘𝑥) = ((rec(𝐹, 𝐴) ↾ ω)‘∅))
18	16, 17	eqeq12d 2246	. . . 4 ⊢ (𝑥 = ∅ → ((frec(𝐹, 𝐴)‘𝑥) = ((rec(𝐹, 𝐴) ↾ ω)‘𝑥) ↔ (frec(𝐹, 𝐴)‘∅) = ((rec(𝐹, 𝐴) ↾ ω)‘∅)))
19		fveq2 5648	. . . . 5 ⊢ (𝑥 = 𝑦 → (frec(𝐹, 𝐴)‘𝑥) = (frec(𝐹, 𝐴)‘𝑦))
20		fveq2 5648	. . . . 5 ⊢ (𝑥 = 𝑦 → ((rec(𝐹, 𝐴) ↾ ω)‘𝑥) = ((rec(𝐹, 𝐴) ↾ ω)‘𝑦))
21	19, 20	eqeq12d 2246	. . . 4 ⊢ (𝑥 = 𝑦 → ((frec(𝐹, 𝐴)‘𝑥) = ((rec(𝐹, 𝐴) ↾ ω)‘𝑥) ↔ (frec(𝐹, 𝐴)‘𝑦) = ((rec(𝐹, 𝐴) ↾ ω)‘𝑦)))
22		fveq2 5648	. . . . 5 ⊢ (𝑥 = suc 𝑦 → (frec(𝐹, 𝐴)‘𝑥) = (frec(𝐹, 𝐴)‘suc 𝑦))
23		fveq2 5648	. . . . 5 ⊢ (𝑥 = suc 𝑦 → ((rec(𝐹, 𝐴) ↾ ω)‘𝑥) = ((rec(𝐹, 𝐴) ↾ ω)‘suc 𝑦))
24	22, 23	eqeq12d 2246	. . . 4 ⊢ (𝑥 = suc 𝑦 → ((frec(𝐹, 𝐴)‘𝑥) = ((rec(𝐹, 𝐴) ↾ ω)‘𝑥) ↔ (frec(𝐹, 𝐴)‘suc 𝑦) = ((rec(𝐹, 𝐴) ↾ ω)‘suc 𝑦)))
25		frec0g 6606	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (frec(𝐹, 𝐴)‘∅) = 𝐴)
26	8, 25	syl 14	. . . . 5 ⊢ (𝜑 → (frec(𝐹, 𝐴)‘∅) = 𝐴)
27		peano1 4698	. . . . . . 7 ⊢ ∅ ∈ ω
28		fvres 5672	. . . . . . 7 ⊢ (∅ ∈ ω → ((rec(𝐹, 𝐴) ↾ ω)‘∅) = (rec(𝐹, 𝐴)‘∅))
29	27, 28	ax-mp 5	. . . . . 6 ⊢ ((rec(𝐹, 𝐴) ↾ ω)‘∅) = (rec(𝐹, 𝐴)‘∅)
30		rdg0g 6597	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → (rec(𝐹, 𝐴)‘∅) = 𝐴)
31	8, 30	syl 14	. . . . . 6 ⊢ (𝜑 → (rec(𝐹, 𝐴)‘∅) = 𝐴)
32	29, 31	eqtrid 2276	. . . . 5 ⊢ (𝜑 → ((rec(𝐹, 𝐴) ↾ ω)‘∅) = 𝐴)
33	26, 32	eqtr4d 2267	. . . 4 ⊢ (𝜑 → (frec(𝐹, 𝐴)‘∅) = ((rec(𝐹, 𝐴) ↾ ω)‘∅))
34		simpr 110	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ω) ∧ (frec(𝐹, 𝐴)‘𝑦) = ((rec(𝐹, 𝐴) ↾ ω)‘𝑦)) → (frec(𝐹, 𝐴)‘𝑦) = ((rec(𝐹, 𝐴) ↾ ω)‘𝑦))
35		fvres 5672	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ω → ((rec(𝐹, 𝐴) ↾ ω)‘𝑦) = (rec(𝐹, 𝐴)‘𝑦))
36	35	ad2antlr 489	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ω) ∧ (frec(𝐹, 𝐴)‘𝑦) = ((rec(𝐹, 𝐴) ↾ ω)‘𝑦)) → ((rec(𝐹, 𝐴) ↾ ω)‘𝑦) = (rec(𝐹, 𝐴)‘𝑦))
37	34, 36	eqtrd 2264	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ ω) ∧ (frec(𝐹, 𝐴)‘𝑦) = ((rec(𝐹, 𝐴) ↾ ω)‘𝑦)) → (frec(𝐹, 𝐴)‘𝑦) = (rec(𝐹, 𝐴)‘𝑦))
38	37	fveq2d 5652	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ ω) ∧ (frec(𝐹, 𝐴)‘𝑦) = ((rec(𝐹, 𝐴) ↾ ω)‘𝑦)) → (𝐹‘(frec(𝐹, 𝐴)‘𝑦)) = (𝐹‘(rec(𝐹, 𝐴)‘𝑦)))
39	7, 8	jca 306	. . . . . . . . . 10 ⊢ (𝜑 → (∀𝑧(𝐹‘𝑧) ∈ V ∧ 𝐴 ∈ 𝑉))
40		simp1 1024	. . . . . . . . . . . . 13 ⊢ ((∀𝑧(𝐹‘𝑧) ∈ V ∧ 𝐴 ∈ 𝑉 ∧ 𝑦 ∈ ω) → ∀𝑧(𝐹‘𝑧) ∈ V)
41		ralv 2821	. . . . . . . . . . . . 13 ⊢ (∀𝑧 ∈ V (𝐹‘𝑧) ∈ V ↔ ∀𝑧(𝐹‘𝑧) ∈ V)
42	40, 41	sylibr 134	. . . . . . . . . . . 12 ⊢ ((∀𝑧(𝐹‘𝑧) ∈ V ∧ 𝐴 ∈ 𝑉 ∧ 𝑦 ∈ ω) → ∀𝑧 ∈ V (𝐹‘𝑧) ∈ V)
43		simp2 1025	. . . . . . . . . . . . 13 ⊢ ((∀𝑧(𝐹‘𝑧) ∈ V ∧ 𝐴 ∈ 𝑉 ∧ 𝑦 ∈ ω) → 𝐴 ∈ 𝑉)
44	43	elexd 2817	. . . . . . . . . . . 12 ⊢ ((∀𝑧(𝐹‘𝑧) ∈ V ∧ 𝐴 ∈ 𝑉 ∧ 𝑦 ∈ ω) → 𝐴 ∈ V)
45		simp3 1026	. . . . . . . . . . . 12 ⊢ ((∀𝑧(𝐹‘𝑧) ∈ V ∧ 𝐴 ∈ 𝑉 ∧ 𝑦 ∈ ω) → 𝑦 ∈ ω)
46		frecsuc 6616	. . . . . . . . . . . 12 ⊢ ((∀𝑧 ∈ V (𝐹‘𝑧) ∈ V ∧ 𝐴 ∈ V ∧ 𝑦 ∈ ω) → (frec(𝐹, 𝐴)‘suc 𝑦) = (𝐹‘(frec(𝐹, 𝐴)‘𝑦)))
47	42, 44, 45, 46	syl3anc 1274	. . . . . . . . . . 11 ⊢ ((∀𝑧(𝐹‘𝑧) ∈ V ∧ 𝐴 ∈ 𝑉 ∧ 𝑦 ∈ ω) → (frec(𝐹, 𝐴)‘suc 𝑦) = (𝐹‘(frec(𝐹, 𝐴)‘𝑦)))
48	47	3expa 1230	. . . . . . . . . 10 ⊢ (((∀𝑧(𝐹‘𝑧) ∈ V ∧ 𝐴 ∈ 𝑉) ∧ 𝑦 ∈ ω) → (frec(𝐹, 𝐴)‘suc 𝑦) = (𝐹‘(frec(𝐹, 𝐴)‘𝑦)))
49	39, 48	sylan 283	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ ω) → (frec(𝐹, 𝐴)‘suc 𝑦) = (𝐹‘(frec(𝐹, 𝐴)‘𝑦)))
50	49	adantr 276	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ ω) ∧ (frec(𝐹, 𝐴)‘𝑦) = ((rec(𝐹, 𝐴) ↾ ω)‘𝑦)) → (frec(𝐹, 𝐴)‘suc 𝑦) = (𝐹‘(frec(𝐹, 𝐴)‘𝑦)))
51	1	adantr 276	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ ω) → 𝐹 Fn V)
52	8	adantr 276	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ ω) → 𝐴 ∈ 𝑉)
53		simpr 110	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ ω) → 𝑦 ∈ ω)
54		nnon 4714	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ω → 𝑦 ∈ On)
55	53, 54	syl 14	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ ω) → 𝑦 ∈ On)
56		frecrdg.inc	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑥 𝑥 ⊆ (𝐹‘𝑥))
57	56	adantr 276	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ ω) → ∀𝑥 𝑥 ⊆ (𝐹‘𝑥))
58	51, 52, 55, 57	rdgisucinc 6594	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ ω) → (rec(𝐹, 𝐴)‘suc 𝑦) = (𝐹‘(rec(𝐹, 𝐴)‘𝑦)))
59	58	adantr 276	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ ω) ∧ (frec(𝐹, 𝐴)‘𝑦) = ((rec(𝐹, 𝐴) ↾ ω)‘𝑦)) → (rec(𝐹, 𝐴)‘suc 𝑦) = (𝐹‘(rec(𝐹, 𝐴)‘𝑦)))
60	38, 50, 59	3eqtr4d 2274	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ ω) ∧ (frec(𝐹, 𝐴)‘𝑦) = ((rec(𝐹, 𝐴) ↾ ω)‘𝑦)) → (frec(𝐹, 𝐴)‘suc 𝑦) = (rec(𝐹, 𝐴)‘suc 𝑦))
61		peano2 4699	. . . . . . . . 9 ⊢ (𝑦 ∈ ω → suc 𝑦 ∈ ω)
62		fvres 5672	. . . . . . . . 9 ⊢ (suc 𝑦 ∈ ω → ((rec(𝐹, 𝐴) ↾ ω)‘suc 𝑦) = (rec(𝐹, 𝐴)‘suc 𝑦))
63	61, 62	syl 14	. . . . . . . 8 ⊢ (𝑦 ∈ ω → ((rec(𝐹, 𝐴) ↾ ω)‘suc 𝑦) = (rec(𝐹, 𝐴)‘suc 𝑦))
64	63	ad2antlr 489	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ ω) ∧ (frec(𝐹, 𝐴)‘𝑦) = ((rec(𝐹, 𝐴) ↾ ω)‘𝑦)) → ((rec(𝐹, 𝐴) ↾ ω)‘suc 𝑦) = (rec(𝐹, 𝐴)‘suc 𝑦))
65	60, 64	eqtr4d 2267	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ ω) ∧ (frec(𝐹, 𝐴)‘𝑦) = ((rec(𝐹, 𝐴) ↾ ω)‘𝑦)) → (frec(𝐹, 𝐴)‘suc 𝑦) = ((rec(𝐹, 𝐴) ↾ ω)‘suc 𝑦))
66	65	ex 115	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ω) → ((frec(𝐹, 𝐴)‘𝑦) = ((rec(𝐹, 𝐴) ↾ ω)‘𝑦) → (frec(𝐹, 𝐴)‘suc 𝑦) = ((rec(𝐹, 𝐴) ↾ ω)‘suc 𝑦)))
67	66	expcom 116	. . . 4 ⊢ (𝑦 ∈ ω → (𝜑 → ((frec(𝐹, 𝐴)‘𝑦) = ((rec(𝐹, 𝐴) ↾ ω)‘𝑦) → (frec(𝐹, 𝐴)‘suc 𝑦) = ((rec(𝐹, 𝐴) ↾ ω)‘suc 𝑦))))
68	18, 21, 24, 33, 67	finds2 4705	. . 3 ⊢ (𝑥 ∈ ω → (𝜑 → (frec(𝐹, 𝐴)‘𝑥) = ((rec(𝐹, 𝐴) ↾ ω)‘𝑥)))
69	68	impcom 125	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ ω) → (frec(𝐹, 𝐴)‘𝑥) = ((rec(𝐹, 𝐴) ↾ ω)‘𝑥))
70	10, 15, 69	eqfnfvd 5756	1 ⊢ (𝜑 → frec(𝐹, 𝐴) = (rec(𝐹, 𝐴) ↾ ω))

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 1005  ∀wal 1396   = wceq 1398   ∈ wcel 2202  ∀wral 2511  Vcvv 2803   ⊆ wss 3201  ∅c0 3496  Oncon0 4466  suc csuc 4468  ωcom 4694   ↾ cres 4733   Fn wfn 5328  ‘cfv 5333  reccrdg 6578  freccfrec 6599

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-iord 4469  df-on 4471  df-ilim 4472  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-recs 6514  df-irdg 6579  df-frec 6600

This theorem is referenced by: (None)