Theorem frecrdg 5931

Description: Transfinite recursion restricted to omega.

Given a suitable characteristic function, df-frec 5918 produces the same results as df-irdg 5897 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 2554	. . . . . 6 ⊢ z ∈ V
3		funfvex 5135	. . . . . . 7 ⊢ ((Fun 𝐹 ∧ z ∈ dom 𝐹) → (𝐹‘z) ∈ V)
4	3	funfni 4942	. . . . . 6 ⊢ ((𝐹 Fn V ∧ z ∈ V) → (𝐹‘z) ∈ V)
5	2, 4	mpan2 401	. . . . 5 ⊢ (𝐹 Fn V → (𝐹‘z) ∈ V)
6	5	alrimiv 1751	. . . 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 5925	. . 3 ⊢ ((∀z(𝐹‘z) ∈ V ∧ A ∈ 𝑉) → frec(𝐹, A) Fn 𝜔)
10	7, 8, 9	syl2anc 391	. 2 ⊢ (φ → frec(𝐹, A) Fn 𝜔)
11		rdgifnon2 5907	. . . 4 ⊢ ((∀z(𝐹‘z) ∈ V ∧ A ∈ 𝑉) → rec(𝐹, A) Fn On)
12	7, 8, 11	syl2anc 391	. . 3 ⊢ (φ → rec(𝐹, A) Fn On)
13		omsson 4278	. . 3 ⊢ 𝜔 ⊆ On
14		fnssres 4955	. . 3 ⊢ ((rec(𝐹, A) Fn On ∧ 𝜔 ⊆ On) → (rec(𝐹, A) ↾ 𝜔) Fn 𝜔)
15	12, 13, 14	sylancl 392	. 2 ⊢ (φ → (rec(𝐹, A) ↾ 𝜔) Fn 𝜔)
16		fveq2 5121	. . . . 5 ⊢ (x = ∅ → (frec(𝐹, A)‘x) = (frec(𝐹, A)‘∅))
17		fveq2 5121	. . . . 5 ⊢ (x = ∅ → ((rec(𝐹, A) ↾ 𝜔)‘x) = ((rec(𝐹, A) ↾ 𝜔)‘∅))
18	16, 17	eqeq12d 2051	. . . 4 ⊢ (x = ∅ → ((frec(𝐹, A)‘x) = ((rec(𝐹, A) ↾ 𝜔)‘x) ↔ (frec(𝐹, A)‘∅) = ((rec(𝐹, A) ↾ 𝜔)‘∅)))
19		fveq2 5121	. . . . 5 ⊢ (x = y → (frec(𝐹, A)‘x) = (frec(𝐹, A)‘y))
20		fveq2 5121	. . . . 5 ⊢ (x = y → ((rec(𝐹, A) ↾ 𝜔)‘x) = ((rec(𝐹, A) ↾ 𝜔)‘y))
21	19, 20	eqeq12d 2051	. . . 4 ⊢ (x = y → ((frec(𝐹, A)‘x) = ((rec(𝐹, A) ↾ 𝜔)‘x) ↔ (frec(𝐹, A)‘y) = ((rec(𝐹, A) ↾ 𝜔)‘y)))
22		fveq2 5121	. . . . 5 ⊢ (x = suc y → (frec(𝐹, A)‘x) = (frec(𝐹, A)‘suc y))
23		fveq2 5121	. . . . 5 ⊢ (x = suc y → ((rec(𝐹, A) ↾ 𝜔)‘x) = ((rec(𝐹, A) ↾ 𝜔)‘suc y))
24	22, 23	eqeq12d 2051	. . . 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 5922	. . . . . 6 ⊢ (A ∈ 𝑉 → (frec(𝐹, A)‘∅) = A)
26	8, 25	syl 14	. . . . 5 ⊢ (φ → (frec(𝐹, A)‘∅) = A)
27		peano1 4260	. . . . . . 7 ⊢ ∅ ∈ 𝜔
28		fvres 5141	. . . . . . 7 ⊢ (∅ ∈ 𝜔 → ((rec(𝐹, A) ↾ 𝜔)‘∅) = (rec(𝐹, A)‘∅))
29	27, 28	ax-mp 7	. . . . . 6 ⊢ ((rec(𝐹, A) ↾ 𝜔)‘∅) = (rec(𝐹, A)‘∅)
30		rdg0g 5915	. . . . . . 7 ⊢ (A ∈ 𝑉 → (rec(𝐹, A)‘∅) = A)
31	8, 30	syl 14	. . . . . 6 ⊢ (φ → (rec(𝐹, A)‘∅) = A)
32	29, 31	syl5eq 2081	. . . . 5 ⊢ (φ → ((rec(𝐹, A) ↾ 𝜔)‘∅) = A)
33	26, 32	eqtr4d 2072	. . . 4 ⊢ (φ → (frec(𝐹, A)‘∅) = ((rec(𝐹, A) ↾ 𝜔)‘∅))
34		simpr 103	. . . . . . . . . 10 ⊢ (((φ ∧ y ∈ 𝜔) ∧ (frec(𝐹, A)‘y) = ((rec(𝐹, A) ↾ 𝜔)‘y)) → (frec(𝐹, A)‘y) = ((rec(𝐹, A) ↾ 𝜔)‘y))
35		fvres 5141	. . . . . . . . . . 11 ⊢ (y ∈ 𝜔 → ((rec(𝐹, A) ↾ 𝜔)‘y) = (rec(𝐹, A)‘y))
36	35	ad2antlr 458	. . . . . . . . . 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 2069	. . . . . . . . 9 ⊢ (((φ ∧ y ∈ 𝜔) ∧ (frec(𝐹, A)‘y) = ((rec(𝐹, A) ↾ 𝜔)‘y)) → (frec(𝐹, A)‘y) = (rec(𝐹, A)‘y))
38	37	fveq2d 5125	. . . . . . . 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 290	. . . . . . . . . 10 ⊢ (φ → (∀z(𝐹‘z) ∈ V ∧ A ∈ 𝑉))
40		frecsuc 5930	. . . . . . . . . . 11 ⊢ ((∀z(𝐹‘z) ∈ V ∧ A ∈ 𝑉 ∧ y ∈ 𝜔) → (frec(𝐹, A)‘suc y) = (𝐹‘(frec(𝐹, A)‘y)))
41	40	3expa 1103	. . . . . . . . . 10 ⊢ (((∀z(𝐹‘z) ∈ V ∧ A ∈ 𝑉) ∧ y ∈ 𝜔) → (frec(𝐹, A)‘suc y) = (𝐹‘(frec(𝐹, A)‘y)))
42	39, 41	sylan 267	. . . . . . . . 9 ⊢ ((φ ∧ y ∈ 𝜔) → (frec(𝐹, A)‘suc y) = (𝐹‘(frec(𝐹, A)‘y)))
43	42	adantr 261	. . . . . . . 8 ⊢ (((φ ∧ y ∈ 𝜔) ∧ (frec(𝐹, A)‘y) = ((rec(𝐹, A) ↾ 𝜔)‘y)) → (frec(𝐹, A)‘suc y) = (𝐹‘(frec(𝐹, A)‘y)))
44	1	adantr 261	. . . . . . . . . 10 ⊢ ((φ ∧ y ∈ 𝜔) → 𝐹 Fn V)
45	8	adantr 261	. . . . . . . . . 10 ⊢ ((φ ∧ y ∈ 𝜔) → A ∈ 𝑉)
46		simpr 103	. . . . . . . . . . 11 ⊢ ((φ ∧ y ∈ 𝜔) → y ∈ 𝜔)
47		nnon 4275	. . . . . . . . . . 11 ⊢ (y ∈ 𝜔 → y ∈ On)
48	46, 47	syl 14	. . . . . . . . . 10 ⊢ ((φ ∧ y ∈ 𝜔) → y ∈ On)
49		frecrdg.inc	. . . . . . . . . . 11 ⊢ (φ → ∀x x ⊆ (𝐹‘x))
50	49	adantr 261	. . . . . . . . . 10 ⊢ ((φ ∧ y ∈ 𝜔) → ∀x x ⊆ (𝐹‘x))
51	44, 45, 48, 50	rdgisucinc 5912	. . . . . . . . 9 ⊢ ((φ ∧ y ∈ 𝜔) → (rec(𝐹, A)‘suc y) = (𝐹‘(rec(𝐹, A)‘y)))
52	51	adantr 261	. . . . . . . 8 ⊢ (((φ ∧ y ∈ 𝜔) ∧ (frec(𝐹, A)‘y) = ((rec(𝐹, A) ↾ 𝜔)‘y)) → (rec(𝐹, A)‘suc y) = (𝐹‘(rec(𝐹, A)‘y)))
53	38, 43, 52	3eqtr4d 2079	. . . . . . 7 ⊢ (((φ ∧ y ∈ 𝜔) ∧ (frec(𝐹, A)‘y) = ((rec(𝐹, A) ↾ 𝜔)‘y)) → (frec(𝐹, A)‘suc y) = (rec(𝐹, A)‘suc y))
54		peano2 4261	. . . . . . . . 9 ⊢ (y ∈ 𝜔 → suc y ∈ 𝜔)
55		fvres 5141	. . . . . . . . 9 ⊢ (suc y ∈ 𝜔 → ((rec(𝐹, A) ↾ 𝜔)‘suc y) = (rec(𝐹, A)‘suc y))
56	54, 55	syl 14	. . . . . . . 8 ⊢ (y ∈ 𝜔 → ((rec(𝐹, A) ↾ 𝜔)‘suc y) = (rec(𝐹, A)‘suc y))
57	56	ad2antlr 458	. . . . . . 7 ⊢ (((φ ∧ y ∈ 𝜔) ∧ (frec(𝐹, A)‘y) = ((rec(𝐹, A) ↾ 𝜔)‘y)) → ((rec(𝐹, A) ↾ 𝜔)‘suc y) = (rec(𝐹, A)‘suc y))
58	53, 57	eqtr4d 2072	. . . . . 6 ⊢ (((φ ∧ y ∈ 𝜔) ∧ (frec(𝐹, A)‘y) = ((rec(𝐹, A) ↾ 𝜔)‘y)) → (frec(𝐹, A)‘suc y) = ((rec(𝐹, A) ↾ 𝜔)‘suc y))
59	58	ex 108	. . . . 5 ⊢ ((φ ∧ y ∈ 𝜔) → ((frec(𝐹, A)‘y) = ((rec(𝐹, A) ↾ 𝜔)‘y) → (frec(𝐹, A)‘suc y) = ((rec(𝐹, A) ↾ 𝜔)‘suc y)))
60	59	expcom 109	. . . 4 ⊢ (y ∈ 𝜔 → (φ → ((frec(𝐹, A)‘y) = ((rec(𝐹, A) ↾ 𝜔)‘y) → (frec(𝐹, A)‘suc y) = ((rec(𝐹, A) ↾ 𝜔)‘suc y))))
61	18, 21, 24, 33, 60	finds2 4267	. . 3 ⊢ (x ∈ 𝜔 → (φ → (frec(𝐹, A)‘x) = ((rec(𝐹, A) ↾ 𝜔)‘x)))
62	61	impcom 116	. 2 ⊢ ((φ ∧ x ∈ 𝜔) → (frec(𝐹, A)‘x) = ((rec(𝐹, A) ↾ 𝜔)‘x))
63	10, 15, 62	eqfnfvd 5211	1 ⊢ (φ → frec(𝐹, A) = (rec(𝐹, A) ↾ 𝜔))

Syntax hints:   → wi 4   ∧ wa 97  ∀wal 1240   = wceq 1242   ∈ wcel 1390  Vcvv 2551   ⊆ wss 2911  ∅c0 3218  Oncon0 4066  suc csuc 4068  𝜔com 4256   ↾ cres 4290   Fn wfn 4840  ‘cfv 4845  reccrdg 5896  freccfrec 5917

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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-iinf 4254

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-tr 3846  df-id 4021  df-iord 4069  df-on 4071  df-suc 4074  df-iom 4257  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-recs 5861  df-irdg 5897  df-frec 5918

This theorem is referenced by: (None)