Theorem frecrdg 6305

Description: Transfinite recursion restricted to omega.

Given a suitable characteristic function, df-frec 6288 produces the same results as df-irdg 6267 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 2689	. . . . . 6 ⊢ 𝑧 ∈ V
3		funfvex 5438	. . . . . . 7 ⊢ ((Fun 𝐹 ∧ 𝑧 ∈ dom 𝐹) → (𝐹‘𝑧) ∈ V)
4	3	funfni 5223	. . . . . 6 ⊢ ((𝐹 Fn V ∧ 𝑧 ∈ V) → (𝐹‘𝑧) ∈ V)
5	2, 4	mpan2 421	. . . . 5 ⊢ (𝐹 Fn V → (𝐹‘𝑧) ∈ V)
6	5	alrimiv 1846	. . . 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 6298	. . 3 ⊢ ((∀𝑧(𝐹‘𝑧) ∈ V ∧ 𝐴 ∈ 𝑉) → frec(𝐹, 𝐴) Fn ω)
10	7, 8, 9	syl2anc 408	. 2 ⊢ (𝜑 → frec(𝐹, 𝐴) Fn ω)
11		rdgifnon2 6277	. . . 4 ⊢ ((∀𝑧(𝐹‘𝑧) ∈ V ∧ 𝐴 ∈ 𝑉) → rec(𝐹, 𝐴) Fn On)
12	7, 8, 11	syl2anc 408	. . 3 ⊢ (𝜑 → rec(𝐹, 𝐴) Fn On)
13		omsson 4526	. . 3 ⊢ ω ⊆ On
14		fnssres 5236	. . 3 ⊢ ((rec(𝐹, 𝐴) Fn On ∧ ω ⊆ On) → (rec(𝐹, 𝐴) ↾ ω) Fn ω)
15	12, 13, 14	sylancl 409	. 2 ⊢ (𝜑 → (rec(𝐹, 𝐴) ↾ ω) Fn ω)
16		fveq2 5421	. . . . 5 ⊢ (𝑥 = ∅ → (frec(𝐹, 𝐴)‘𝑥) = (frec(𝐹, 𝐴)‘∅))
17		fveq2 5421	. . . . 5 ⊢ (𝑥 = ∅ → ((rec(𝐹, 𝐴) ↾ ω)‘𝑥) = ((rec(𝐹, 𝐴) ↾ ω)‘∅))
18	16, 17	eqeq12d 2154	. . . 4 ⊢ (𝑥 = ∅ → ((frec(𝐹, 𝐴)‘𝑥) = ((rec(𝐹, 𝐴) ↾ ω)‘𝑥) ↔ (frec(𝐹, 𝐴)‘∅) = ((rec(𝐹, 𝐴) ↾ ω)‘∅)))
19		fveq2 5421	. . . . 5 ⊢ (𝑥 = 𝑦 → (frec(𝐹, 𝐴)‘𝑥) = (frec(𝐹, 𝐴)‘𝑦))
20		fveq2 5421	. . . . 5 ⊢ (𝑥 = 𝑦 → ((rec(𝐹, 𝐴) ↾ ω)‘𝑥) = ((rec(𝐹, 𝐴) ↾ ω)‘𝑦))
21	19, 20	eqeq12d 2154	. . . 4 ⊢ (𝑥 = 𝑦 → ((frec(𝐹, 𝐴)‘𝑥) = ((rec(𝐹, 𝐴) ↾ ω)‘𝑥) ↔ (frec(𝐹, 𝐴)‘𝑦) = ((rec(𝐹, 𝐴) ↾ ω)‘𝑦)))
22		fveq2 5421	. . . . 5 ⊢ (𝑥 = suc 𝑦 → (frec(𝐹, 𝐴)‘𝑥) = (frec(𝐹, 𝐴)‘suc 𝑦))
23		fveq2 5421	. . . . 5 ⊢ (𝑥 = suc 𝑦 → ((rec(𝐹, 𝐴) ↾ ω)‘𝑥) = ((rec(𝐹, 𝐴) ↾ ω)‘suc 𝑦))
24	22, 23	eqeq12d 2154	. . . 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 6294	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (frec(𝐹, 𝐴)‘∅) = 𝐴)
26	8, 25	syl 14	. . . . 5 ⊢ (𝜑 → (frec(𝐹, 𝐴)‘∅) = 𝐴)
27		peano1 4508	. . . . . . 7 ⊢ ∅ ∈ ω
28		fvres 5445	. . . . . . 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 6285	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → (rec(𝐹, 𝐴)‘∅) = 𝐴)
31	8, 30	syl 14	. . . . . 6 ⊢ (𝜑 → (rec(𝐹, 𝐴)‘∅) = 𝐴)
32	29, 31	syl5eq 2184	. . . . 5 ⊢ (𝜑 → ((rec(𝐹, 𝐴) ↾ ω)‘∅) = 𝐴)
33	26, 32	eqtr4d 2175	. . . 4 ⊢ (𝜑 → (frec(𝐹, 𝐴)‘∅) = ((rec(𝐹, 𝐴) ↾ ω)‘∅))
34		simpr 109	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ω) ∧ (frec(𝐹, 𝐴)‘𝑦) = ((rec(𝐹, 𝐴) ↾ ω)‘𝑦)) → (frec(𝐹, 𝐴)‘𝑦) = ((rec(𝐹, 𝐴) ↾ ω)‘𝑦))
35		fvres 5445	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ω → ((rec(𝐹, 𝐴) ↾ ω)‘𝑦) = (rec(𝐹, 𝐴)‘𝑦))
36	35	ad2antlr 480	. . . . . . . . . 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 2172	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ ω) ∧ (frec(𝐹, 𝐴)‘𝑦) = ((rec(𝐹, 𝐴) ↾ ω)‘𝑦)) → (frec(𝐹, 𝐴)‘𝑦) = (rec(𝐹, 𝐴)‘𝑦))
38	37	fveq2d 5425	. . . . . . . 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 304	. . . . . . . . . 10 ⊢ (𝜑 → (∀𝑧(𝐹‘𝑧) ∈ V ∧ 𝐴 ∈ 𝑉))
40		simp1 981	. . . . . . . . . . . . 13 ⊢ ((∀𝑧(𝐹‘𝑧) ∈ V ∧ 𝐴 ∈ 𝑉 ∧ 𝑦 ∈ ω) → ∀𝑧(𝐹‘𝑧) ∈ V)
41		ralv 2703	. . . . . . . . . . . . 13 ⊢ (∀𝑧 ∈ V (𝐹‘𝑧) ∈ V ↔ ∀𝑧(𝐹‘𝑧) ∈ V)
42	40, 41	sylibr 133	. . . . . . . . . . . 12 ⊢ ((∀𝑧(𝐹‘𝑧) ∈ V ∧ 𝐴 ∈ 𝑉 ∧ 𝑦 ∈ ω) → ∀𝑧 ∈ V (𝐹‘𝑧) ∈ V)
43		simp2 982	. . . . . . . . . . . . 13 ⊢ ((∀𝑧(𝐹‘𝑧) ∈ V ∧ 𝐴 ∈ 𝑉 ∧ 𝑦 ∈ ω) → 𝐴 ∈ 𝑉)
44	43	elexd 2699	. . . . . . . . . . . 12 ⊢ ((∀𝑧(𝐹‘𝑧) ∈ V ∧ 𝐴 ∈ 𝑉 ∧ 𝑦 ∈ ω) → 𝐴 ∈ V)
45		simp3 983	. . . . . . . . . . . 12 ⊢ ((∀𝑧(𝐹‘𝑧) ∈ V ∧ 𝐴 ∈ 𝑉 ∧ 𝑦 ∈ ω) → 𝑦 ∈ ω)
46		frecsuc 6304	. . . . . . . . . . . 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 1216	. . . . . . . . . . 11 ⊢ ((∀𝑧(𝐹‘𝑧) ∈ V ∧ 𝐴 ∈ 𝑉 ∧ 𝑦 ∈ ω) → (frec(𝐹, 𝐴)‘suc 𝑦) = (𝐹‘(frec(𝐹, 𝐴)‘𝑦)))
48	47	3expa 1181	. . . . . . . . . 10 ⊢ (((∀𝑧(𝐹‘𝑧) ∈ V ∧ 𝐴 ∈ 𝑉) ∧ 𝑦 ∈ ω) → (frec(𝐹, 𝐴)‘suc 𝑦) = (𝐹‘(frec(𝐹, 𝐴)‘𝑦)))
49	39, 48	sylan 281	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ ω) → (frec(𝐹, 𝐴)‘suc 𝑦) = (𝐹‘(frec(𝐹, 𝐴)‘𝑦)))
50	49	adantr 274	. . . . . . . 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 274	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ ω) → 𝐹 Fn V)
52	8	adantr 274	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ ω) → 𝐴 ∈ 𝑉)
53		simpr 109	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ ω) → 𝑦 ∈ ω)
54		nnon 4523	. . . . . . . . . . 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 274	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ ω) → ∀𝑥 𝑥 ⊆ (𝐹‘𝑥))
58	51, 52, 55, 57	rdgisucinc 6282	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ ω) → (rec(𝐹, 𝐴)‘suc 𝑦) = (𝐹‘(rec(𝐹, 𝐴)‘𝑦)))
59	58	adantr 274	. . . . . . . 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 2182	. . . . . . 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 4509	. . . . . . . . 9 ⊢ (𝑦 ∈ ω → suc 𝑦 ∈ ω)
62		fvres 5445	. . . . . . . . 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 480	. . . . . . 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 2175	. . . . . 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 114	. . . . 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 115	. . . 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 4515	. . 3 ⊢ (𝑥 ∈ ω → (𝜑 → (frec(𝐹, 𝐴)‘𝑥) = ((rec(𝐹, 𝐴) ↾ ω)‘𝑥)))
69	68	impcom 124	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ ω) → (frec(𝐹, 𝐴)‘𝑥) = ((rec(𝐹, 𝐴) ↾ ω)‘𝑥))
70	10, 15, 69	eqfnfvd 5521	1 ⊢ (𝜑 → frec(𝐹, 𝐴) = (rec(𝐹, 𝐴) ↾ ω))

Syntax hints:   → wi 4   ∧ wa 103   ∧ w3a 962  ∀wal 1329   = wceq 1331   ∈ wcel 1480  ∀wral 2416  Vcvv 2686   ⊆ wss 3071  ∅c0 3363  Oncon0 4285  suc csuc 4287  ωcom 4504   ↾ cres 4541   Fn wfn 5118  ‘cfv 5123  reccrdg 6266  freccfrec 6287

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-iord 4288  df-on 4290  df-ilim 4291  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-recs 6202  df-irdg 6267  df-frec 6288

This theorem is referenced by: (None)