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Theorem frecrdg 6409
Description: Transfinite recursion restricted to omega.

Given a suitable characteristic function, df-frec 6392 produces the same results as df-irdg 6371 restricted to ω.

Presumably the theorem would also hold if 𝐹 Fn V were changed to 𝑧(𝐹𝑧) ∈ 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 2741 . . . . . 6 𝑧 ∈ V
3 funfvex 5533 . . . . . . 7 ((Fun 𝐹𝑧 ∈ dom 𝐹) → (𝐹𝑧) ∈ V)
43funfni 5317 . . . . . 6 ((𝐹 Fn V ∧ 𝑧 ∈ V) → (𝐹𝑧) ∈ V)
52, 4mpan2 425 . . . . 5 (𝐹 Fn V → (𝐹𝑧) ∈ V)
65alrimiv 1874 . . . 4 (𝐹 Fn V → ∀𝑧(𝐹𝑧) ∈ V)
71, 6syl 14 . . 3 (𝜑 → ∀𝑧(𝐹𝑧) ∈ V)
8 frecrdg.2 . . 3 (𝜑𝐴𝑉)
9 frecfnom 6402 . . 3 ((∀𝑧(𝐹𝑧) ∈ V ∧ 𝐴𝑉) → frec(𝐹, 𝐴) Fn ω)
107, 8, 9syl2anc 411 . 2 (𝜑 → frec(𝐹, 𝐴) Fn ω)
11 rdgifnon2 6381 . . . 4 ((∀𝑧(𝐹𝑧) ∈ V ∧ 𝐴𝑉) → rec(𝐹, 𝐴) Fn On)
127, 8, 11syl2anc 411 . . 3 (𝜑 → rec(𝐹, 𝐴) Fn On)
13 omsson 4613 . . 3 ω ⊆ On
14 fnssres 5330 . . 3 ((rec(𝐹, 𝐴) Fn On ∧ ω ⊆ On) → (rec(𝐹, 𝐴) ↾ ω) Fn ω)
1512, 13, 14sylancl 413 . 2 (𝜑 → (rec(𝐹, 𝐴) ↾ ω) Fn ω)
16 fveq2 5516 . . . . 5 (𝑥 = ∅ → (frec(𝐹, 𝐴)‘𝑥) = (frec(𝐹, 𝐴)‘∅))
17 fveq2 5516 . . . . 5 (𝑥 = ∅ → ((rec(𝐹, 𝐴) ↾ ω)‘𝑥) = ((rec(𝐹, 𝐴) ↾ ω)‘∅))
1816, 17eqeq12d 2192 . . . 4 (𝑥 = ∅ → ((frec(𝐹, 𝐴)‘𝑥) = ((rec(𝐹, 𝐴) ↾ ω)‘𝑥) ↔ (frec(𝐹, 𝐴)‘∅) = ((rec(𝐹, 𝐴) ↾ ω)‘∅)))
19 fveq2 5516 . . . . 5 (𝑥 = 𝑦 → (frec(𝐹, 𝐴)‘𝑥) = (frec(𝐹, 𝐴)‘𝑦))
20 fveq2 5516 . . . . 5 (𝑥 = 𝑦 → ((rec(𝐹, 𝐴) ↾ ω)‘𝑥) = ((rec(𝐹, 𝐴) ↾ ω)‘𝑦))
2119, 20eqeq12d 2192 . . . 4 (𝑥 = 𝑦 → ((frec(𝐹, 𝐴)‘𝑥) = ((rec(𝐹, 𝐴) ↾ ω)‘𝑥) ↔ (frec(𝐹, 𝐴)‘𝑦) = ((rec(𝐹, 𝐴) ↾ ω)‘𝑦)))
22 fveq2 5516 . . . . 5 (𝑥 = suc 𝑦 → (frec(𝐹, 𝐴)‘𝑥) = (frec(𝐹, 𝐴)‘suc 𝑦))
23 fveq2 5516 . . . . 5 (𝑥 = suc 𝑦 → ((rec(𝐹, 𝐴) ↾ ω)‘𝑥) = ((rec(𝐹, 𝐴) ↾ ω)‘suc 𝑦))
2422, 23eqeq12d 2192 . . . 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 6398 . . . . . 6 (𝐴𝑉 → (frec(𝐹, 𝐴)‘∅) = 𝐴)
268, 25syl 14 . . . . 5 (𝜑 → (frec(𝐹, 𝐴)‘∅) = 𝐴)
27 peano1 4594 . . . . . . 7 ∅ ∈ ω
28 fvres 5540 . . . . . . 7 (∅ ∈ ω → ((rec(𝐹, 𝐴) ↾ ω)‘∅) = (rec(𝐹, 𝐴)‘∅))
2927, 28ax-mp 5 . . . . . 6 ((rec(𝐹, 𝐴) ↾ ω)‘∅) = (rec(𝐹, 𝐴)‘∅)
30 rdg0g 6389 . . . . . . 7 (𝐴𝑉 → (rec(𝐹, 𝐴)‘∅) = 𝐴)
318, 30syl 14 . . . . . 6 (𝜑 → (rec(𝐹, 𝐴)‘∅) = 𝐴)
3229, 31eqtrid 2222 . . . . 5 (𝜑 → ((rec(𝐹, 𝐴) ↾ ω)‘∅) = 𝐴)
3326, 32eqtr4d 2213 . . . 4 (𝜑 → (frec(𝐹, 𝐴)‘∅) = ((rec(𝐹, 𝐴) ↾ ω)‘∅))
34 simpr 110 . . . . . . . . . 10 (((𝜑𝑦 ∈ ω) ∧ (frec(𝐹, 𝐴)‘𝑦) = ((rec(𝐹, 𝐴) ↾ ω)‘𝑦)) → (frec(𝐹, 𝐴)‘𝑦) = ((rec(𝐹, 𝐴) ↾ ω)‘𝑦))
35 fvres 5540 . . . . . . . . . . 11 (𝑦 ∈ ω → ((rec(𝐹, 𝐴) ↾ ω)‘𝑦) = (rec(𝐹, 𝐴)‘𝑦))
3635ad2antlr 489 . . . . . . . . . 10 (((𝜑𝑦 ∈ ω) ∧ (frec(𝐹, 𝐴)‘𝑦) = ((rec(𝐹, 𝐴) ↾ ω)‘𝑦)) → ((rec(𝐹, 𝐴) ↾ ω)‘𝑦) = (rec(𝐹, 𝐴)‘𝑦))
3734, 36eqtrd 2210 . . . . . . . . 9 (((𝜑𝑦 ∈ ω) ∧ (frec(𝐹, 𝐴)‘𝑦) = ((rec(𝐹, 𝐴) ↾ ω)‘𝑦)) → (frec(𝐹, 𝐴)‘𝑦) = (rec(𝐹, 𝐴)‘𝑦))
3837fveq2d 5520 . . . . . . . 8 (((𝜑𝑦 ∈ ω) ∧ (frec(𝐹, 𝐴)‘𝑦) = ((rec(𝐹, 𝐴) ↾ ω)‘𝑦)) → (𝐹‘(frec(𝐹, 𝐴)‘𝑦)) = (𝐹‘(rec(𝐹, 𝐴)‘𝑦)))
397, 8jca 306 . . . . . . . . . 10 (𝜑 → (∀𝑧(𝐹𝑧) ∈ V ∧ 𝐴𝑉))
40 simp1 997 . . . . . . . . . . . . 13 ((∀𝑧(𝐹𝑧) ∈ V ∧ 𝐴𝑉𝑦 ∈ ω) → ∀𝑧(𝐹𝑧) ∈ V)
41 ralv 2755 . . . . . . . . . . . . 13 (∀𝑧 ∈ V (𝐹𝑧) ∈ V ↔ ∀𝑧(𝐹𝑧) ∈ V)
4240, 41sylibr 134 . . . . . . . . . . . 12 ((∀𝑧(𝐹𝑧) ∈ V ∧ 𝐴𝑉𝑦 ∈ ω) → ∀𝑧 ∈ V (𝐹𝑧) ∈ V)
43 simp2 998 . . . . . . . . . . . . 13 ((∀𝑧(𝐹𝑧) ∈ V ∧ 𝐴𝑉𝑦 ∈ ω) → 𝐴𝑉)
4443elexd 2751 . . . . . . . . . . . 12 ((∀𝑧(𝐹𝑧) ∈ V ∧ 𝐴𝑉𝑦 ∈ ω) → 𝐴 ∈ V)
45 simp3 999 . . . . . . . . . . . 12 ((∀𝑧(𝐹𝑧) ∈ V ∧ 𝐴𝑉𝑦 ∈ ω) → 𝑦 ∈ ω)
46 frecsuc 6408 . . . . . . . . . . . 12 ((∀𝑧 ∈ V (𝐹𝑧) ∈ V ∧ 𝐴 ∈ V ∧ 𝑦 ∈ ω) → (frec(𝐹, 𝐴)‘suc 𝑦) = (𝐹‘(frec(𝐹, 𝐴)‘𝑦)))
4742, 44, 45, 46syl3anc 1238 . . . . . . . . . . 11 ((∀𝑧(𝐹𝑧) ∈ V ∧ 𝐴𝑉𝑦 ∈ ω) → (frec(𝐹, 𝐴)‘suc 𝑦) = (𝐹‘(frec(𝐹, 𝐴)‘𝑦)))
48473expa 1203 . . . . . . . . . 10 (((∀𝑧(𝐹𝑧) ∈ V ∧ 𝐴𝑉) ∧ 𝑦 ∈ ω) → (frec(𝐹, 𝐴)‘suc 𝑦) = (𝐹‘(frec(𝐹, 𝐴)‘𝑦)))
4939, 48sylan 283 . . . . . . . . 9 ((𝜑𝑦 ∈ ω) → (frec(𝐹, 𝐴)‘suc 𝑦) = (𝐹‘(frec(𝐹, 𝐴)‘𝑦)))
5049adantr 276 . . . . . . . 8 (((𝜑𝑦 ∈ ω) ∧ (frec(𝐹, 𝐴)‘𝑦) = ((rec(𝐹, 𝐴) ↾ ω)‘𝑦)) → (frec(𝐹, 𝐴)‘suc 𝑦) = (𝐹‘(frec(𝐹, 𝐴)‘𝑦)))
511adantr 276 . . . . . . . . . 10 ((𝜑𝑦 ∈ ω) → 𝐹 Fn V)
528adantr 276 . . . . . . . . . 10 ((𝜑𝑦 ∈ ω) → 𝐴𝑉)
53 simpr 110 . . . . . . . . . . 11 ((𝜑𝑦 ∈ ω) → 𝑦 ∈ ω)
54 nnon 4610 . . . . . . . . . . 11 (𝑦 ∈ ω → 𝑦 ∈ On)
5553, 54syl 14 . . . . . . . . . 10 ((𝜑𝑦 ∈ ω) → 𝑦 ∈ On)
56 frecrdg.inc . . . . . . . . . . 11 (𝜑 → ∀𝑥 𝑥 ⊆ (𝐹𝑥))
5756adantr 276 . . . . . . . . . 10 ((𝜑𝑦 ∈ ω) → ∀𝑥 𝑥 ⊆ (𝐹𝑥))
5851, 52, 55, 57rdgisucinc 6386 . . . . . . . . 9 ((𝜑𝑦 ∈ ω) → (rec(𝐹, 𝐴)‘suc 𝑦) = (𝐹‘(rec(𝐹, 𝐴)‘𝑦)))
5958adantr 276 . . . . . . . 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 2220 . . . . . . 7 (((𝜑𝑦 ∈ ω) ∧ (frec(𝐹, 𝐴)‘𝑦) = ((rec(𝐹, 𝐴) ↾ ω)‘𝑦)) → (frec(𝐹, 𝐴)‘suc 𝑦) = (rec(𝐹, 𝐴)‘suc 𝑦))
61 peano2 4595 . . . . . . . . 9 (𝑦 ∈ ω → suc 𝑦 ∈ ω)
62 fvres 5540 . . . . . . . . 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 489 . . . . . . 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 2213 . . . . . 6 (((𝜑𝑦 ∈ ω) ∧ (frec(𝐹, 𝐴)‘𝑦) = ((rec(𝐹, 𝐴) ↾ ω)‘𝑦)) → (frec(𝐹, 𝐴)‘suc 𝑦) = ((rec(𝐹, 𝐴) ↾ ω)‘suc 𝑦))
6665ex 115 . . . . 5 ((𝜑𝑦 ∈ ω) → ((frec(𝐹, 𝐴)‘𝑦) = ((rec(𝐹, 𝐴) ↾ ω)‘𝑦) → (frec(𝐹, 𝐴)‘suc 𝑦) = ((rec(𝐹, 𝐴) ↾ ω)‘suc 𝑦)))
6766expcom 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))))
6818, 21, 24, 33, 67finds2 4601 . . 3 (𝑥 ∈ ω → (𝜑 → (frec(𝐹, 𝐴)‘𝑥) = ((rec(𝐹, 𝐴) ↾ ω)‘𝑥)))
6968impcom 125 . 2 ((𝜑𝑥 ∈ ω) → (frec(𝐹, 𝐴)‘𝑥) = ((rec(𝐹, 𝐴) ↾ ω)‘𝑥))
7010, 15, 69eqfnfvd 5617 1 (𝜑 → frec(𝐹, 𝐴) = (rec(𝐹, 𝐴) ↾ ω))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  w3a 978  wal 1351   = wceq 1353  wcel 2148  wral 2455  Vcvv 2738  wss 3130  c0 3423  Oncon0 4364  suc csuc 4366  ωcom 4590  cres 4629   Fn wfn 5212  cfv 5217  reccrdg 6370  freccfrec 6391
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4119  ax-sep 4122  ax-nul 4130  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-iinf 4588
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-tr 4103  df-id 4294  df-iord 4367  df-on 4369  df-ilim 4370  df-suc 4372  df-iom 4591  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-recs 6306  df-irdg 6371  df-frec 6392
This theorem is referenced by: (None)
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