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Theorem frecrdg 6408
Description: Transfinite recursion restricted to omega.

Given a suitable characteristic function, df-frec 6391 produces the same results as df-irdg 6370 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 2740 . . . . . 6 𝑧 ∈ V
3 funfvex 5532 . . . . . . 7 ((Fun 𝐹𝑧 ∈ dom 𝐹) → (𝐹𝑧) ∈ V)
43funfni 5316 . . . . . 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 6401 . . 3 ((∀𝑧(𝐹𝑧) ∈ V ∧ 𝐴𝑉) → frec(𝐹, 𝐴) Fn ω)
107, 8, 9syl2anc 411 . 2 (𝜑 → frec(𝐹, 𝐴) Fn ω)
11 rdgifnon2 6380 . . . 4 ((∀𝑧(𝐹𝑧) ∈ V ∧ 𝐴𝑉) → rec(𝐹, 𝐴) Fn On)
127, 8, 11syl2anc 411 . . 3 (𝜑 → rec(𝐹, 𝐴) Fn On)
13 omsson 4612 . . 3 ω ⊆ On
14 fnssres 5329 . . 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 5515 . . . . 5 (𝑥 = ∅ → (frec(𝐹, 𝐴)‘𝑥) = (frec(𝐹, 𝐴)‘∅))
17 fveq2 5515 . . . . 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 5515 . . . . 5 (𝑥 = 𝑦 → (frec(𝐹, 𝐴)‘𝑥) = (frec(𝐹, 𝐴)‘𝑦))
20 fveq2 5515 . . . . 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 5515 . . . . 5 (𝑥 = suc 𝑦 → (frec(𝐹, 𝐴)‘𝑥) = (frec(𝐹, 𝐴)‘suc 𝑦))
23 fveq2 5515 . . . . 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 6397 . . . . . 6 (𝐴𝑉 → (frec(𝐹, 𝐴)‘∅) = 𝐴)
268, 25syl 14 . . . . 5 (𝜑 → (frec(𝐹, 𝐴)‘∅) = 𝐴)
27 peano1 4593 . . . . . . 7 ∅ ∈ ω
28 fvres 5539 . . . . . . 7 (∅ ∈ ω → ((rec(𝐹, 𝐴) ↾ ω)‘∅) = (rec(𝐹, 𝐴)‘∅))
2927, 28ax-mp 5 . . . . . 6 ((rec(𝐹, 𝐴) ↾ ω)‘∅) = (rec(𝐹, 𝐴)‘∅)
30 rdg0g 6388 . . . . . . 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 5539 . . . . . . . . . . 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 5519 . . . . . . . 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 2754 . . . . . . . . . . . . 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 2750 . . . . . . . . . . . 12 ((∀𝑧(𝐹𝑧) ∈ V ∧ 𝐴𝑉𝑦 ∈ ω) → 𝐴 ∈ V)
45 simp3 999 . . . . . . . . . . . 12 ((∀𝑧(𝐹𝑧) ∈ V ∧ 𝐴𝑉𝑦 ∈ ω) → 𝑦 ∈ ω)
46 frecsuc 6407 . . . . . . . . . . . 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 4609 . . . . . . . . . . 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 6385 . . . . . . . . 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 4594 . . . . . . . . 9 (𝑦 ∈ ω → suc 𝑦 ∈ ω)
62 fvres 5539 . . . . . . . . 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 4600 . . 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 5616 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 2737  wss 3129  c0 3422  Oncon0 4363  suc csuc 4365  ωcom 4589  cres 4628   Fn wfn 5211  cfv 5216  reccrdg 6369  freccfrec 6390
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 4118  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-iinf 4587
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 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-tr 4102  df-id 4293  df-iord 4366  df-on 4368  df-ilim 4369  df-suc 4371  df-iom 4590  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-recs 6305  df-irdg 6370  df-frec 6391
This theorem is referenced by: (None)
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