Theorem frecrdg 6669

Description: Transfinite recursion restricted to omega.

Given a suitable characteristic function, df-frec 6652 produces the same results as df-irdg 6631 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 2824	. . . . . 6 ⊢ 𝑧 ∈ V
3		funfvex 5707	. . . . . . 7 ⊢ ((Fun 𝐹 ∧ 𝑧 ∈ dom 𝐹) → (𝐹‘𝑧) ∈ V)
4	3	funfni 5478	. . . . . 6 ⊢ ((𝐹 Fn V ∧ 𝑧 ∈ V) → (𝐹‘𝑧) ∈ V)
5	2, 4	mpan2 429	. . . . 5 ⊢ (𝐹 Fn V → (𝐹‘𝑧) ∈ V)
6	5	alrimiv 1927	. . . 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 6662	. . 3 ⊢ ((∀𝑧(𝐹‘𝑧) ∈ V ∧ 𝐴 ∈ 𝑉) → frec(𝐹, 𝐴) Fn ω)
10	7, 8, 9	syl2anc 415	. 2 ⊢ (𝜑 → frec(𝐹, 𝐴) Fn ω)
11		rdgifnon2 6641	. . . 4 ⊢ ((∀𝑧(𝐹‘𝑧) ∈ V ∧ 𝐴 ∈ 𝑉) → rec(𝐹, 𝐴) Fn On)
12	7, 8, 11	syl2anc 415	. . 3 ⊢ (𝜑 → rec(𝐹, 𝐴) Fn On)
13		omsson 4755	. . 3 ⊢ ω ⊆ On
14		fnssres 5491	. . 3 ⊢ ((rec(𝐹, 𝐴) Fn On ∧ ω ⊆ On) → (rec(𝐹, 𝐴) ↾ ω) Fn ω)
15	12, 13, 14	sylancl 417	. 2 ⊢ (𝜑 → (rec(𝐹, 𝐴) ↾ ω) Fn ω)
16		fveq2 5690	. . . . 5 ⊢ (𝑥 = ∅ → (frec(𝐹, 𝐴)‘𝑥) = (frec(𝐹, 𝐴)‘∅))
17		fveq2 5690	. . . . 5 ⊢ (𝑥 = ∅ → ((rec(𝐹, 𝐴) ↾ ω)‘𝑥) = ((rec(𝐹, 𝐴) ↾ ω)‘∅))
18	16, 17	eqeq12d 2253	. . . 4 ⊢ (𝑥 = ∅ → ((frec(𝐹, 𝐴)‘𝑥) = ((rec(𝐹, 𝐴) ↾ ω)‘𝑥) ↔ (frec(𝐹, 𝐴)‘∅) = ((rec(𝐹, 𝐴) ↾ ω)‘∅)))
19		fveq2 5690	. . . . 5 ⊢ (𝑥 = 𝑦 → (frec(𝐹, 𝐴)‘𝑥) = (frec(𝐹, 𝐴)‘𝑦))
20		fveq2 5690	. . . . 5 ⊢ (𝑥 = 𝑦 → ((rec(𝐹, 𝐴) ↾ ω)‘𝑥) = ((rec(𝐹, 𝐴) ↾ ω)‘𝑦))
21	19, 20	eqeq12d 2253	. . . 4 ⊢ (𝑥 = 𝑦 → ((frec(𝐹, 𝐴)‘𝑥) = ((rec(𝐹, 𝐴) ↾ ω)‘𝑥) ↔ (frec(𝐹, 𝐴)‘𝑦) = ((rec(𝐹, 𝐴) ↾ ω)‘𝑦)))
22		fveq2 5690	. . . . 5 ⊢ (𝑥 = suc 𝑦 → (frec(𝐹, 𝐴)‘𝑥) = (frec(𝐹, 𝐴)‘suc 𝑦))
23		fveq2 5690	. . . . 5 ⊢ (𝑥 = suc 𝑦 → ((rec(𝐹, 𝐴) ↾ ω)‘𝑥) = ((rec(𝐹, 𝐴) ↾ ω)‘suc 𝑦))
24	22, 23	eqeq12d 2253	. . . 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 6658	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (frec(𝐹, 𝐴)‘∅) = 𝐴)
26	8, 25	syl 14	. . . . 5 ⊢ (𝜑 → (frec(𝐹, 𝐴)‘∅) = 𝐴)
27		peano1 4736	. . . . . . 7 ⊢ ∅ ∈ ω
28		fvres 5714	. . . . . . 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 6649	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → (rec(𝐹, 𝐴)‘∅) = 𝐴)
31	8, 30	syl 14	. . . . . 6 ⊢ (𝜑 → (rec(𝐹, 𝐴)‘∅) = 𝐴)
32	29, 31	eqtrid 2283	. . . . 5 ⊢ (𝜑 → ((rec(𝐹, 𝐴) ↾ ω)‘∅) = 𝐴)
33	26, 32	eqtr4d 2274	. . . 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 5714	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ω → ((rec(𝐹, 𝐴) ↾ ω)‘𝑦) = (rec(𝐹, 𝐴)‘𝑦))
36	35	ad2antlr 493	. . . . . . . . . 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 2271	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ ω) ∧ (frec(𝐹, 𝐴)‘𝑦) = ((rec(𝐹, 𝐴) ↾ ω)‘𝑦)) → (frec(𝐹, 𝐴)‘𝑦) = (rec(𝐹, 𝐴)‘𝑦))
38	37	fveq2d 5694	. . . . . . . 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 1028	. . . . . . . . . . . . 13 ⊢ ((∀𝑧(𝐹‘𝑧) ∈ V ∧ 𝐴 ∈ 𝑉 ∧ 𝑦 ∈ ω) → ∀𝑧(𝐹‘𝑧) ∈ V)
41		ralv 2839	. . . . . . . . . . . . 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 1029	. . . . . . . . . . . . 13 ⊢ ((∀𝑧(𝐹‘𝑧) ∈ V ∧ 𝐴 ∈ 𝑉 ∧ 𝑦 ∈ ω) → 𝐴 ∈ 𝑉)
44	43	elexd 2835	. . . . . . . . . . . 12 ⊢ ((∀𝑧(𝐹‘𝑧) ∈ V ∧ 𝐴 ∈ 𝑉 ∧ 𝑦 ∈ ω) → 𝐴 ∈ V)
45		simp3 1030	. . . . . . . . . . . 12 ⊢ ((∀𝑧(𝐹‘𝑧) ∈ V ∧ 𝐴 ∈ 𝑉 ∧ 𝑦 ∈ ω) → 𝑦 ∈ ω)
46		frecsuc 6668	. . . . . . . . . . . 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 1278	. . . . . . . . . . 11 ⊢ ((∀𝑧(𝐹‘𝑧) ∈ V ∧ 𝐴 ∈ 𝑉 ∧ 𝑦 ∈ ω) → (frec(𝐹, 𝐴)‘suc 𝑦) = (𝐹‘(frec(𝐹, 𝐴)‘𝑦)))
48	47	3expa 1234	. . . . . . . . . 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 4752	. . . . . . . . . . 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 6646	. . . . . . . . 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 2281	. . . . . . 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 4737	. . . . . . . . 9 ⊢ (𝑦 ∈ ω → suc 𝑦 ∈ ω)
62		fvres 5714	. . . . . . . . 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 493	. . . . . . 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 2274	. . . . . 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 4743	. . 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 5800	1 ⊢ (𝜑 → frec(𝐹, 𝐴) = (rec(𝐹, 𝐴) ↾ ω))

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 1009  ∀wal 1400   = wceq 1402   ∈ wcel 2209  ∀wral 2528  Vcvv 2821   ⊆ wss 3220  ∅c0 3520  Oncon0 4503  suc csuc 4505  ωcom 4732   ↾ cres 4771   Fn wfn 5367  ‘cfv 5372  reccrdg 6630  freccfrec 6651

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-iord 4506  df-on 4508  df-ilim 4509  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-recs 6566  df-irdg 6631  df-frec 6652

This theorem is referenced by: (None)