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Theorem frecrdg 6023
Description: Transfinite recursion restricted to omega.

Given a suitable characteristic function, df-frec 6009 produces the same results as df-irdg 5988 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 2577 . . . . . 6 𝑧 ∈ V
3 funfvex 5220 . . . . . . 7 ((Fun 𝐹𝑧 ∈ dom 𝐹) → (𝐹𝑧) ∈ V)
43funfni 5027 . . . . . 6 ((𝐹 Fn V ∧ 𝑧 ∈ V) → (𝐹𝑧) ∈ V)
52, 4mpan2 409 . . . . 5 (𝐹 Fn V → (𝐹𝑧) ∈ V)
65alrimiv 1770 . . . 4 (𝐹 Fn V → ∀𝑧(𝐹𝑧) ∈ V)
71, 6syl 14 . . 3 (𝜑 → ∀𝑧(𝐹𝑧) ∈ V)
8 frecrdg.2 . . 3 (𝜑𝐴𝑉)
9 frecfnom 6017 . . 3 ((∀𝑧(𝐹𝑧) ∈ V ∧ 𝐴𝑉) → frec(𝐹, 𝐴) Fn ω)
107, 8, 9syl2anc 397 . 2 (𝜑 → frec(𝐹, 𝐴) Fn ω)
11 rdgifnon2 5998 . . . 4 ((∀𝑧(𝐹𝑧) ∈ V ∧ 𝐴𝑉) → rec(𝐹, 𝐴) Fn On)
127, 8, 11syl2anc 397 . . 3 (𝜑 → rec(𝐹, 𝐴) Fn On)
13 omsson 4363 . . 3 ω ⊆ On
14 fnssres 5040 . . 3 ((rec(𝐹, 𝐴) Fn On ∧ ω ⊆ On) → (rec(𝐹, 𝐴) ↾ ω) Fn ω)
1512, 13, 14sylancl 398 . 2 (𝜑 → (rec(𝐹, 𝐴) ↾ ω) Fn ω)
16 fveq2 5206 . . . . 5 (𝑥 = ∅ → (frec(𝐹, 𝐴)‘𝑥) = (frec(𝐹, 𝐴)‘∅))
17 fveq2 5206 . . . . 5 (𝑥 = ∅ → ((rec(𝐹, 𝐴) ↾ ω)‘𝑥) = ((rec(𝐹, 𝐴) ↾ ω)‘∅))
1816, 17eqeq12d 2070 . . . 4 (𝑥 = ∅ → ((frec(𝐹, 𝐴)‘𝑥) = ((rec(𝐹, 𝐴) ↾ ω)‘𝑥) ↔ (frec(𝐹, 𝐴)‘∅) = ((rec(𝐹, 𝐴) ↾ ω)‘∅)))
19 fveq2 5206 . . . . 5 (𝑥 = 𝑦 → (frec(𝐹, 𝐴)‘𝑥) = (frec(𝐹, 𝐴)‘𝑦))
20 fveq2 5206 . . . . 5 (𝑥 = 𝑦 → ((rec(𝐹, 𝐴) ↾ ω)‘𝑥) = ((rec(𝐹, 𝐴) ↾ ω)‘𝑦))
2119, 20eqeq12d 2070 . . . 4 (𝑥 = 𝑦 → ((frec(𝐹, 𝐴)‘𝑥) = ((rec(𝐹, 𝐴) ↾ ω)‘𝑥) ↔ (frec(𝐹, 𝐴)‘𝑦) = ((rec(𝐹, 𝐴) ↾ ω)‘𝑦)))
22 fveq2 5206 . . . . 5 (𝑥 = suc 𝑦 → (frec(𝐹, 𝐴)‘𝑥) = (frec(𝐹, 𝐴)‘suc 𝑦))
23 fveq2 5206 . . . . 5 (𝑥 = suc 𝑦 → ((rec(𝐹, 𝐴) ↾ ω)‘𝑥) = ((rec(𝐹, 𝐴) ↾ ω)‘suc 𝑦))
2422, 23eqeq12d 2070 . . . 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 6014 . . . . . 6 (𝐴𝑉 → (frec(𝐹, 𝐴)‘∅) = 𝐴)
268, 25syl 14 . . . . 5 (𝜑 → (frec(𝐹, 𝐴)‘∅) = 𝐴)
27 peano1 4345 . . . . . . 7 ∅ ∈ ω
28 fvres 5226 . . . . . . 7 (∅ ∈ ω → ((rec(𝐹, 𝐴) ↾ ω)‘∅) = (rec(𝐹, 𝐴)‘∅))
2927, 28ax-mp 7 . . . . . 6 ((rec(𝐹, 𝐴) ↾ ω)‘∅) = (rec(𝐹, 𝐴)‘∅)
30 rdg0g 6006 . . . . . . 7 (𝐴𝑉 → (rec(𝐹, 𝐴)‘∅) = 𝐴)
318, 30syl 14 . . . . . 6 (𝜑 → (rec(𝐹, 𝐴)‘∅) = 𝐴)
3229, 31syl5eq 2100 . . . . 5 (𝜑 → ((rec(𝐹, 𝐴) ↾ ω)‘∅) = 𝐴)
3326, 32eqtr4d 2091 . . . 4 (𝜑 → (frec(𝐹, 𝐴)‘∅) = ((rec(𝐹, 𝐴) ↾ ω)‘∅))
34 simpr 107 . . . . . . . . . 10 (((𝜑𝑦 ∈ ω) ∧ (frec(𝐹, 𝐴)‘𝑦) = ((rec(𝐹, 𝐴) ↾ ω)‘𝑦)) → (frec(𝐹, 𝐴)‘𝑦) = ((rec(𝐹, 𝐴) ↾ ω)‘𝑦))
35 fvres 5226 . . . . . . . . . . 11 (𝑦 ∈ ω → ((rec(𝐹, 𝐴) ↾ ω)‘𝑦) = (rec(𝐹, 𝐴)‘𝑦))
3635ad2antlr 466 . . . . . . . . . 10 (((𝜑𝑦 ∈ ω) ∧ (frec(𝐹, 𝐴)‘𝑦) = ((rec(𝐹, 𝐴) ↾ ω)‘𝑦)) → ((rec(𝐹, 𝐴) ↾ ω)‘𝑦) = (rec(𝐹, 𝐴)‘𝑦))
3734, 36eqtrd 2088 . . . . . . . . 9 (((𝜑𝑦 ∈ ω) ∧ (frec(𝐹, 𝐴)‘𝑦) = ((rec(𝐹, 𝐴) ↾ ω)‘𝑦)) → (frec(𝐹, 𝐴)‘𝑦) = (rec(𝐹, 𝐴)‘𝑦))
3837fveq2d 5210 . . . . . . . 8 (((𝜑𝑦 ∈ ω) ∧ (frec(𝐹, 𝐴)‘𝑦) = ((rec(𝐹, 𝐴) ↾ ω)‘𝑦)) → (𝐹‘(frec(𝐹, 𝐴)‘𝑦)) = (𝐹‘(rec(𝐹, 𝐴)‘𝑦)))
397, 8jca 294 . . . . . . . . . 10 (𝜑 → (∀𝑧(𝐹𝑧) ∈ V ∧ 𝐴𝑉))
40 frecsuc 6022 . . . . . . . . . . 11 ((∀𝑧(𝐹𝑧) ∈ V ∧ 𝐴𝑉𝑦 ∈ ω) → (frec(𝐹, 𝐴)‘suc 𝑦) = (𝐹‘(frec(𝐹, 𝐴)‘𝑦)))
41403expa 1115 . . . . . . . . . 10 (((∀𝑧(𝐹𝑧) ∈ V ∧ 𝐴𝑉) ∧ 𝑦 ∈ ω) → (frec(𝐹, 𝐴)‘suc 𝑦) = (𝐹‘(frec(𝐹, 𝐴)‘𝑦)))
4239, 41sylan 271 . . . . . . . . 9 ((𝜑𝑦 ∈ ω) → (frec(𝐹, 𝐴)‘suc 𝑦) = (𝐹‘(frec(𝐹, 𝐴)‘𝑦)))
4342adantr 265 . . . . . . . 8 (((𝜑𝑦 ∈ ω) ∧ (frec(𝐹, 𝐴)‘𝑦) = ((rec(𝐹, 𝐴) ↾ ω)‘𝑦)) → (frec(𝐹, 𝐴)‘suc 𝑦) = (𝐹‘(frec(𝐹, 𝐴)‘𝑦)))
441adantr 265 . . . . . . . . . 10 ((𝜑𝑦 ∈ ω) → 𝐹 Fn V)
458adantr 265 . . . . . . . . . 10 ((𝜑𝑦 ∈ ω) → 𝐴𝑉)
46 simpr 107 . . . . . . . . . . 11 ((𝜑𝑦 ∈ ω) → 𝑦 ∈ ω)
47 nnon 4360 . . . . . . . . . . 11 (𝑦 ∈ ω → 𝑦 ∈ On)
4846, 47syl 14 . . . . . . . . . 10 ((𝜑𝑦 ∈ ω) → 𝑦 ∈ On)
49 frecrdg.inc . . . . . . . . . . 11 (𝜑 → ∀𝑥 𝑥 ⊆ (𝐹𝑥))
5049adantr 265 . . . . . . . . . 10 ((𝜑𝑦 ∈ ω) → ∀𝑥 𝑥 ⊆ (𝐹𝑥))
5144, 45, 48, 50rdgisucinc 6003 . . . . . . . . 9 ((𝜑𝑦 ∈ ω) → (rec(𝐹, 𝐴)‘suc 𝑦) = (𝐹‘(rec(𝐹, 𝐴)‘𝑦)))
5251adantr 265 . . . . . . . 8 (((𝜑𝑦 ∈ ω) ∧ (frec(𝐹, 𝐴)‘𝑦) = ((rec(𝐹, 𝐴) ↾ ω)‘𝑦)) → (rec(𝐹, 𝐴)‘suc 𝑦) = (𝐹‘(rec(𝐹, 𝐴)‘𝑦)))
5338, 43, 523eqtr4d 2098 . . . . . . 7 (((𝜑𝑦 ∈ ω) ∧ (frec(𝐹, 𝐴)‘𝑦) = ((rec(𝐹, 𝐴) ↾ ω)‘𝑦)) → (frec(𝐹, 𝐴)‘suc 𝑦) = (rec(𝐹, 𝐴)‘suc 𝑦))
54 peano2 4346 . . . . . . . . 9 (𝑦 ∈ ω → suc 𝑦 ∈ ω)
55 fvres 5226 . . . . . . . . 9 (suc 𝑦 ∈ ω → ((rec(𝐹, 𝐴) ↾ ω)‘suc 𝑦) = (rec(𝐹, 𝐴)‘suc 𝑦))
5654, 55syl 14 . . . . . . . 8 (𝑦 ∈ ω → ((rec(𝐹, 𝐴) ↾ ω)‘suc 𝑦) = (rec(𝐹, 𝐴)‘suc 𝑦))
5756ad2antlr 466 . . . . . . 7 (((𝜑𝑦 ∈ ω) ∧ (frec(𝐹, 𝐴)‘𝑦) = ((rec(𝐹, 𝐴) ↾ ω)‘𝑦)) → ((rec(𝐹, 𝐴) ↾ ω)‘suc 𝑦) = (rec(𝐹, 𝐴)‘suc 𝑦))
5853, 57eqtr4d 2091 . . . . . 6 (((𝜑𝑦 ∈ ω) ∧ (frec(𝐹, 𝐴)‘𝑦) = ((rec(𝐹, 𝐴) ↾ ω)‘𝑦)) → (frec(𝐹, 𝐴)‘suc 𝑦) = ((rec(𝐹, 𝐴) ↾ ω)‘suc 𝑦))
5958ex 112 . . . . 5 ((𝜑𝑦 ∈ ω) → ((frec(𝐹, 𝐴)‘𝑦) = ((rec(𝐹, 𝐴) ↾ ω)‘𝑦) → (frec(𝐹, 𝐴)‘suc 𝑦) = ((rec(𝐹, 𝐴) ↾ ω)‘suc 𝑦)))
6059expcom 113 . . . 4 (𝑦 ∈ ω → (𝜑 → ((frec(𝐹, 𝐴)‘𝑦) = ((rec(𝐹, 𝐴) ↾ ω)‘𝑦) → (frec(𝐹, 𝐴)‘suc 𝑦) = ((rec(𝐹, 𝐴) ↾ ω)‘suc 𝑦))))
6118, 21, 24, 33, 60finds2 4352 . . 3 (𝑥 ∈ ω → (𝜑 → (frec(𝐹, 𝐴)‘𝑥) = ((rec(𝐹, 𝐴) ↾ ω)‘𝑥)))
6261impcom 120 . 2 ((𝜑𝑥 ∈ ω) → (frec(𝐹, 𝐴)‘𝑥) = ((rec(𝐹, 𝐴) ↾ ω)‘𝑥))
6310, 15, 62eqfnfvd 5296 1 (𝜑 → frec(𝐹, 𝐴) = (rec(𝐹, 𝐴) ↾ ω))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wal 1257   = wceq 1259  wcel 1409  Vcvv 2574  wss 2945  c0 3252  Oncon0 4128  suc csuc 4130  ωcom 4341  cres 4375   Fn wfn 4925  cfv 4930  reccrdg 5987  freccfrec 6008
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3900  ax-sep 3903  ax-nul 3911  ax-pow 3955  ax-pr 3972  ax-un 4198  ax-setind 4290  ax-iinf 4339
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2788  df-csb 2881  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-nul 3253  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-int 3644  df-iun 3687  df-br 3793  df-opab 3847  df-mpt 3848  df-tr 3883  df-id 4058  df-iord 4131  df-on 4133  df-suc 4136  df-iom 4342  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-f1 4935  df-fo 4936  df-f1o 4937  df-fv 4938  df-recs 5951  df-irdg 5988  df-frec 6009
This theorem is referenced by: (None)
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