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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 (𝜑𝐹 Fn V)
frecrdg.2 (𝜑𝐴𝑉)
frecrdg.inc (𝜑 → ∀𝑥 𝑥 ⊆ (𝐹𝑥))
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.
StepHypRef Expression
1 frecrdg.1 . . . 4 (𝜑𝐹 Fn V)
2 vex 2689 . . . . . 6 𝑧 ∈ V
3 funfvex 5438 . . . . . . 7 ((Fun 𝐹𝑧 ∈ dom 𝐹) → (𝐹𝑧) ∈ V)
43funfni 5223 . . . . . 6 ((𝐹 Fn V ∧ 𝑧 ∈ V) → (𝐹𝑧) ∈ V)
52, 4mpan2 421 . . . . 5 (𝐹 Fn V → (𝐹𝑧) ∈ V)
65alrimiv 1846 . . . 4 (𝐹 Fn V → ∀𝑧(𝐹𝑧) ∈ V)
71, 6syl 14 . . 3 (𝜑 → ∀𝑧(𝐹𝑧) ∈ V)
8 frecrdg.2 . . 3 (𝜑𝐴𝑉)
9 frecfnom 6298 . . 3 ((∀𝑧(𝐹𝑧) ∈ V ∧ 𝐴𝑉) → frec(𝐹, 𝐴) Fn ω)
107, 8, 9syl2anc 408 . 2 (𝜑 → frec(𝐹, 𝐴) Fn ω)
11 rdgifnon2 6277 . . . 4 ((∀𝑧(𝐹𝑧) ∈ V ∧ 𝐴𝑉) → rec(𝐹, 𝐴) Fn On)
127, 8, 11syl2anc 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)
1512, 13, 14sylancl 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)‘∅))
1816, 17eqeq12d 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))
2119, 20eqeq12d 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))
2422, 23eqeq12d 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 (𝐴𝑉 → (frec(𝐹, 𝐴)‘∅) = 𝐴)
268, 25syl 14 . . . . 5 (𝜑 → (frec(𝐹, 𝐴)‘∅) = 𝐴)
27 peano1 4508 . . . . . . 7 ∅ ∈ ω
28 fvres 5445 . . . . . . 7 (∅ ∈ ω → ((rec(𝐹, 𝐴) ↾ ω)‘∅) = (rec(𝐹, 𝐴)‘∅))
2927, 28ax-mp 5 . . . . . 6 ((rec(𝐹, 𝐴) ↾ ω)‘∅) = (rec(𝐹, 𝐴)‘∅)
30 rdg0g 6285 . . . . . . 7 (𝐴𝑉 → (rec(𝐹, 𝐴)‘∅) = 𝐴)
318, 30syl 14 . . . . . 6 (𝜑 → (rec(𝐹, 𝐴)‘∅) = 𝐴)
3229, 31syl5eq 2184 . . . . 5 (𝜑 → ((rec(𝐹, 𝐴) ↾ ω)‘∅) = 𝐴)
3326, 32eqtr4d 2175 . . . 4 (𝜑 → (frec(𝐹, 𝐴)‘∅) = ((rec(𝐹, 𝐴) ↾ ω)‘∅))
34 simpr 109 . . . . . . . . . 10 (((𝜑𝑦 ∈ 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))
3635ad2antlr 480 . . . . . . . . . 10 (((𝜑𝑦 ∈ ω) ∧ (frec(𝐹, 𝐴)‘𝑦) = ((rec(𝐹, 𝐴) ↾ ω)‘𝑦)) → ((rec(𝐹, 𝐴) ↾ ω)‘𝑦) = (rec(𝐹, 𝐴)‘𝑦))
3734, 36eqtrd 2172 . . . . . . . . 9 (((𝜑𝑦 ∈ ω) ∧ (frec(𝐹, 𝐴)‘𝑦) = ((rec(𝐹, 𝐴) ↾ ω)‘𝑦)) → (frec(𝐹, 𝐴)‘𝑦) = (rec(𝐹, 𝐴)‘𝑦))
3837fveq2d 5425 . . . . . . . 8 (((𝜑𝑦 ∈ ω) ∧ (frec(𝐹, 𝐴)‘𝑦) = ((rec(𝐹, 𝐴) ↾ ω)‘𝑦)) → (𝐹‘(frec(𝐹, 𝐴)‘𝑦)) = (𝐹‘(rec(𝐹, 𝐴)‘𝑦)))
397, 8jca 304 . . . . . . . . . 10 (𝜑 → (∀𝑧(𝐹𝑧) ∈ V ∧ 𝐴𝑉))
40 simp1 981 . . . . . . . . . . . . 13 ((∀𝑧(𝐹𝑧) ∈ V ∧ 𝐴𝑉𝑦 ∈ ω) → ∀𝑧(𝐹𝑧) ∈ V)
41 ralv 2703 . . . . . . . . . . . . 13 (∀𝑧 ∈ V (𝐹𝑧) ∈ V ↔ ∀𝑧(𝐹𝑧) ∈ V)
4240, 41sylibr 133 . . . . . . . . . . . 12 ((∀𝑧(𝐹𝑧) ∈ V ∧ 𝐴𝑉𝑦 ∈ ω) → ∀𝑧 ∈ V (𝐹𝑧) ∈ V)
43 simp2 982 . . . . . . . . . . . . 13 ((∀𝑧(𝐹𝑧) ∈ V ∧ 𝐴𝑉𝑦 ∈ ω) → 𝐴𝑉)
4443elexd 2699 . . . . . . . . . . . 12 ((∀𝑧(𝐹𝑧) ∈ V ∧ 𝐴𝑉𝑦 ∈ ω) → 𝐴 ∈ V)
45 simp3 983 . . . . . . . . . . . 12 ((∀𝑧(𝐹𝑧) ∈ V ∧ 𝐴𝑉𝑦 ∈ ω) → 𝑦 ∈ ω)
46 frecsuc 6304 . . . . . . . . . . . 12 ((∀𝑧 ∈ V (𝐹𝑧) ∈ V ∧ 𝐴 ∈ V ∧ 𝑦 ∈ ω) → (frec(𝐹, 𝐴)‘suc 𝑦) = (𝐹‘(frec(𝐹, 𝐴)‘𝑦)))
4742, 44, 45, 46syl3anc 1216 . . . . . . . . . . 11 ((∀𝑧(𝐹𝑧) ∈ V ∧ 𝐴𝑉𝑦 ∈ ω) → (frec(𝐹, 𝐴)‘suc 𝑦) = (𝐹‘(frec(𝐹, 𝐴)‘𝑦)))
48473expa 1181 . . . . . . . . . 10 (((∀𝑧(𝐹𝑧) ∈ V ∧ 𝐴𝑉) ∧ 𝑦 ∈ ω) → (frec(𝐹, 𝐴)‘suc 𝑦) = (𝐹‘(frec(𝐹, 𝐴)‘𝑦)))
4939, 48sylan 281 . . . . . . . . 9 ((𝜑𝑦 ∈ ω) → (frec(𝐹, 𝐴)‘suc 𝑦) = (𝐹‘(frec(𝐹, 𝐴)‘𝑦)))
5049adantr 274 . . . . . . . 8 (((𝜑𝑦 ∈ ω) ∧ (frec(𝐹, 𝐴)‘𝑦) = ((rec(𝐹, 𝐴) ↾ ω)‘𝑦)) → (frec(𝐹, 𝐴)‘suc 𝑦) = (𝐹‘(frec(𝐹, 𝐴)‘𝑦)))
511adantr 274 . . . . . . . . . 10 ((𝜑𝑦 ∈ ω) → 𝐹 Fn V)
528adantr 274 . . . . . . . . . 10 ((𝜑𝑦 ∈ ω) → 𝐴𝑉)
53 simpr 109 . . . . . . . . . . 11 ((𝜑𝑦 ∈ ω) → 𝑦 ∈ ω)
54 nnon 4523 . . . . . . . . . . 11 (𝑦 ∈ ω → 𝑦 ∈ On)
5553, 54syl 14 . . . . . . . . . 10 ((𝜑𝑦 ∈ ω) → 𝑦 ∈ On)
56 frecrdg.inc . . . . . . . . . . 11 (𝜑 → ∀𝑥 𝑥 ⊆ (𝐹𝑥))
5756adantr 274 . . . . . . . . . 10 ((𝜑𝑦 ∈ ω) → ∀𝑥 𝑥 ⊆ (𝐹𝑥))
5851, 52, 55, 57rdgisucinc 6282 . . . . . . . . 9 ((𝜑𝑦 ∈ ω) → (rec(𝐹, 𝐴)‘suc 𝑦) = (𝐹‘(rec(𝐹, 𝐴)‘𝑦)))
5958adantr 274 . . . . . . . 8 (((𝜑𝑦 ∈ ω) ∧ (frec(𝐹, 𝐴)‘𝑦) = ((rec(𝐹, 𝐴) ↾ ω)‘𝑦)) → (rec(𝐹, 𝐴)‘suc 𝑦) = (𝐹‘(rec(𝐹, 𝐴)‘𝑦)))
6038, 50, 593eqtr4d 2182 . . . . . . 7 (((𝜑𝑦 ∈ 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))
6361, 62syl 14 . . . . . . . 8 (𝑦 ∈ ω → ((rec(𝐹, 𝐴) ↾ ω)‘suc 𝑦) = (rec(𝐹, 𝐴)‘suc 𝑦))
6463ad2antlr 480 . . . . . . 7 (((𝜑𝑦 ∈ ω) ∧ (frec(𝐹, 𝐴)‘𝑦) = ((rec(𝐹, 𝐴) ↾ ω)‘𝑦)) → ((rec(𝐹, 𝐴) ↾ ω)‘suc 𝑦) = (rec(𝐹, 𝐴)‘suc 𝑦))
6560, 64eqtr4d 2175 . . . . . 6 (((𝜑𝑦 ∈ ω) ∧ (frec(𝐹, 𝐴)‘𝑦) = ((rec(𝐹, 𝐴) ↾ ω)‘𝑦)) → (frec(𝐹, 𝐴)‘suc 𝑦) = ((rec(𝐹, 𝐴) ↾ ω)‘suc 𝑦))
6665ex 114 . . . . 5 ((𝜑𝑦 ∈ ω) → ((frec(𝐹, 𝐴)‘𝑦) = ((rec(𝐹, 𝐴) ↾ ω)‘𝑦) → (frec(𝐹, 𝐴)‘suc 𝑦) = ((rec(𝐹, 𝐴) ↾ ω)‘suc 𝑦)))
6766expcom 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))))
6818, 21, 24, 33, 67finds2 4515 . . 3 (𝑥 ∈ ω → (𝜑 → (frec(𝐹, 𝐴)‘𝑥) = ((rec(𝐹, 𝐴) ↾ ω)‘𝑥)))
6968impcom 124 . 2 ((𝜑𝑥 ∈ ω) → (frec(𝐹, 𝐴)‘𝑥) = ((rec(𝐹, 𝐴) ↾ ω)‘𝑥))
7010, 15, 69eqfnfvd 5521 1 (𝜑 → frec(𝐹, 𝐴) = (rec(𝐹, 𝐴) ↾ ω))
 Colors of variables: wff set class 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)
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