Theorem frecrdg 6507

Description: Transfinite recursion restricted to omega.

Given a suitable characteristic function, df-frec 6490 produces the same results as df-irdg 6469 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 2776	. . . . . 6 ⊢ 𝑧 ∈ V
3		funfvex 5606	. . . . . . 7 ⊢ ((Fun 𝐹 ∧ 𝑧 ∈ dom 𝐹) → (𝐹‘𝑧) ∈ V)
4	3	funfni 5385	. . . . . 6 ⊢ ((𝐹 Fn V ∧ 𝑧 ∈ V) → (𝐹‘𝑧) ∈ V)
5	2, 4	mpan2 425	. . . . 5 ⊢ (𝐹 Fn V → (𝐹‘𝑧) ∈ V)
6	5	alrimiv 1898	. . . 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 6500	. . 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 6479	. . . 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 4669	. . 3 ⊢ ω ⊆ On
14		fnssres 5398	. . 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 5589	. . . . 5 ⊢ (𝑥 = ∅ → (frec(𝐹, 𝐴)‘𝑥) = (frec(𝐹, 𝐴)‘∅))
17		fveq2 5589	. . . . 5 ⊢ (𝑥 = ∅ → ((rec(𝐹, 𝐴) ↾ ω)‘𝑥) = ((rec(𝐹, 𝐴) ↾ ω)‘∅))
18	16, 17	eqeq12d 2221	. . . 4 ⊢ (𝑥 = ∅ → ((frec(𝐹, 𝐴)‘𝑥) = ((rec(𝐹, 𝐴) ↾ ω)‘𝑥) ↔ (frec(𝐹, 𝐴)‘∅) = ((rec(𝐹, 𝐴) ↾ ω)‘∅)))
19		fveq2 5589	. . . . 5 ⊢ (𝑥 = 𝑦 → (frec(𝐹, 𝐴)‘𝑥) = (frec(𝐹, 𝐴)‘𝑦))
20		fveq2 5589	. . . . 5 ⊢ (𝑥 = 𝑦 → ((rec(𝐹, 𝐴) ↾ ω)‘𝑥) = ((rec(𝐹, 𝐴) ↾ ω)‘𝑦))
21	19, 20	eqeq12d 2221	. . . 4 ⊢ (𝑥 = 𝑦 → ((frec(𝐹, 𝐴)‘𝑥) = ((rec(𝐹, 𝐴) ↾ ω)‘𝑥) ↔ (frec(𝐹, 𝐴)‘𝑦) = ((rec(𝐹, 𝐴) ↾ ω)‘𝑦)))
22		fveq2 5589	. . . . 5 ⊢ (𝑥 = suc 𝑦 → (frec(𝐹, 𝐴)‘𝑥) = (frec(𝐹, 𝐴)‘suc 𝑦))
23		fveq2 5589	. . . . 5 ⊢ (𝑥 = suc 𝑦 → ((rec(𝐹, 𝐴) ↾ ω)‘𝑥) = ((rec(𝐹, 𝐴) ↾ ω)‘suc 𝑦))
24	22, 23	eqeq12d 2221	. . . 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 6496	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (frec(𝐹, 𝐴)‘∅) = 𝐴)
26	8, 25	syl 14	. . . . 5 ⊢ (𝜑 → (frec(𝐹, 𝐴)‘∅) = 𝐴)
27		peano1 4650	. . . . . . 7 ⊢ ∅ ∈ ω
28		fvres 5613	. . . . . . 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 6487	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → (rec(𝐹, 𝐴)‘∅) = 𝐴)
31	8, 30	syl 14	. . . . . 6 ⊢ (𝜑 → (rec(𝐹, 𝐴)‘∅) = 𝐴)
32	29, 31	eqtrid 2251	. . . . 5 ⊢ (𝜑 → ((rec(𝐹, 𝐴) ↾ ω)‘∅) = 𝐴)
33	26, 32	eqtr4d 2242	. . . 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 5613	. . . . . . . . . . 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 2239	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ ω) ∧ (frec(𝐹, 𝐴)‘𝑦) = ((rec(𝐹, 𝐴) ↾ ω)‘𝑦)) → (frec(𝐹, 𝐴)‘𝑦) = (rec(𝐹, 𝐴)‘𝑦))
38	37	fveq2d 5593	. . . . . . . 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 1000	. . . . . . . . . . . . 13 ⊢ ((∀𝑧(𝐹‘𝑧) ∈ V ∧ 𝐴 ∈ 𝑉 ∧ 𝑦 ∈ ω) → ∀𝑧(𝐹‘𝑧) ∈ V)
41		ralv 2791	. . . . . . . . . . . . 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 1001	. . . . . . . . . . . . 13 ⊢ ((∀𝑧(𝐹‘𝑧) ∈ V ∧ 𝐴 ∈ 𝑉 ∧ 𝑦 ∈ ω) → 𝐴 ∈ 𝑉)
44	43	elexd 2787	. . . . . . . . . . . 12 ⊢ ((∀𝑧(𝐹‘𝑧) ∈ V ∧ 𝐴 ∈ 𝑉 ∧ 𝑦 ∈ ω) → 𝐴 ∈ V)
45		simp3 1002	. . . . . . . . . . . 12 ⊢ ((∀𝑧(𝐹‘𝑧) ∈ V ∧ 𝐴 ∈ 𝑉 ∧ 𝑦 ∈ ω) → 𝑦 ∈ ω)
46		frecsuc 6506	. . . . . . . . . . . 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 1250	. . . . . . . . . . 11 ⊢ ((∀𝑧(𝐹‘𝑧) ∈ V ∧ 𝐴 ∈ 𝑉 ∧ 𝑦 ∈ ω) → (frec(𝐹, 𝐴)‘suc 𝑦) = (𝐹‘(frec(𝐹, 𝐴)‘𝑦)))
48	47	3expa 1206	. . . . . . . . . 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 4666	. . . . . . . . . . 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 6484	. . . . . . . . 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 2249	. . . . . . 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 4651	. . . . . . . . 9 ⊢ (𝑦 ∈ ω → suc 𝑦 ∈ ω)
62		fvres 5613	. . . . . . . . 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 2242	. . . . . 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 4657	. . 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 5693	1 ⊢ (𝜑 → frec(𝐹, 𝐴) = (rec(𝐹, 𝐴) ↾ ω))

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 981  ∀wal 1371   = wceq 1373   ∈ wcel 2177  ∀wral 2485  Vcvv 2773   ⊆ wss 3170  ∅c0 3464  Oncon0 4418  suc csuc 4420  ωcom 4646   ↾ cres 4685   Fn wfn 5275  ‘cfv 5280  reccrdg 6468  freccfrec 6489

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4167  ax-sep 4170  ax-nul 4178  ax-pow 4226  ax-pr 4261  ax-un 4488  ax-setind 4593  ax-iinf 4644

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-ral 2490  df-rex 2491  df-reu 2492  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3857  df-int 3892  df-iun 3935  df-br 4052  df-opab 4114  df-mpt 4115  df-tr 4151  df-id 4348  df-iord 4421  df-on 4423  df-ilim 4424  df-suc 4426  df-iom 4647  df-xp 4689  df-rel 4690  df-cnv 4691  df-co 4692  df-dm 4693  df-rn 4694  df-res 4695  df-ima 4696  df-iota 5241  df-fun 5282  df-fn 5283  df-f 5284  df-f1 5285  df-fo 5286  df-f1o 5287  df-fv 5288  df-recs 6404  df-irdg 6469  df-frec 6490

This theorem is referenced by: (None)