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Theorem nffrec 6300
 Description: Bound-variable hypothesis builder for the finite recursive definition generator. (Contributed by Jim Kingdon, 30-May-2020.)
Hypotheses
Ref Expression
nffrec.1 𝑥𝐹
nffrec.2 𝑥𝐴
Assertion
Ref Expression
nffrec 𝑥frec(𝐹, 𝐴)

Proof of Theorem nffrec
Dummy variables 𝑔 𝑚 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-frec 6295 . 2 frec(𝐹, 𝐴) = (recs((𝑔 ∈ V ↦ {𝑦 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑦 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑦𝐴))})) ↾ ω)
2 nfcv 2282 . . . . 5 𝑥V
3 nfcv 2282 . . . . . . . 8 𝑥ω
4 nfv 1509 . . . . . . . . 9 𝑥dom 𝑔 = suc 𝑚
5 nffrec.1 . . . . . . . . . . 11 𝑥𝐹
6 nfcv 2282 . . . . . . . . . . 11 𝑥(𝑔𝑚)
75, 6nffv 5438 . . . . . . . . . 10 𝑥(𝐹‘(𝑔𝑚))
87nfcri 2276 . . . . . . . . 9 𝑥 𝑦 ∈ (𝐹‘(𝑔𝑚))
94, 8nfan 1545 . . . . . . . 8 𝑥(dom 𝑔 = suc 𝑚𝑦 ∈ (𝐹‘(𝑔𝑚)))
103, 9nfrexya 2477 . . . . . . 7 𝑥𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑦 ∈ (𝐹‘(𝑔𝑚)))
11 nfv 1509 . . . . . . . 8 𝑥dom 𝑔 = ∅
12 nffrec.2 . . . . . . . . 9 𝑥𝐴
1312nfcri 2276 . . . . . . . 8 𝑥 𝑦𝐴
1411, 13nfan 1545 . . . . . . 7 𝑥(dom 𝑔 = ∅ ∧ 𝑦𝐴)
1510, 14nfor 1554 . . . . . 6 𝑥(∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑦 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑦𝐴))
1615nfab 2287 . . . . 5 𝑥{𝑦 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑦 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑦𝐴))}
172, 16nfmpt 4027 . . . 4 𝑥(𝑔 ∈ V ↦ {𝑦 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑦 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑦𝐴))})
1817nfrecs 6211 . . 3 𝑥recs((𝑔 ∈ V ↦ {𝑦 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑦 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑦𝐴))}))
1918, 3nfres 4828 . 2 𝑥(recs((𝑔 ∈ V ↦ {𝑦 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑦 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑦𝐴))})) ↾ ω)
201, 19nfcxfr 2279 1 𝑥frec(𝐹, 𝐴)
 Colors of variables: wff set class Syntax hints:   ∧ wa 103   ∨ wo 698   = wceq 1332   ∈ wcel 1481  {cab 2126  Ⅎwnfc 2269  ∃wrex 2418  Vcvv 2689  ∅c0 3367   ↦ cmpt 3996  suc csuc 4294  ωcom 4511  dom cdm 4546   ↾ cres 4548  ‘cfv 5130  recscrecs 6208  freccfrec 6294 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-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-rab 2426  df-v 2691  df-un 3079  df-in 3081  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-br 3937  df-opab 3997  df-mpt 3998  df-xp 4552  df-res 4558  df-iota 5095  df-fv 5138  df-recs 6209  df-frec 6295 This theorem is referenced by:  nfseq  10258
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