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Theorem nffrec 6045
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 6040 . 2 frec(𝐹, 𝐴) = (recs((𝑔 ∈ V ↦ {𝑦 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑦 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑦𝐴))})) ↾ ω)
2 nfcv 2220 . . . . 5 𝑥V
3 nfcv 2220 . . . . . . . 8 𝑥ω
4 nfv 1462 . . . . . . . . 9 𝑥dom 𝑔 = suc 𝑚
5 nffrec.1 . . . . . . . . . . 11 𝑥𝐹
6 nfcv 2220 . . . . . . . . . . 11 𝑥(𝑔𝑚)
75, 6nffv 5216 . . . . . . . . . 10 𝑥(𝐹‘(𝑔𝑚))
87nfcri 2214 . . . . . . . . 9 𝑥 𝑦 ∈ (𝐹‘(𝑔𝑚))
94, 8nfan 1498 . . . . . . . 8 𝑥(dom 𝑔 = suc 𝑚𝑦 ∈ (𝐹‘(𝑔𝑚)))
103, 9nfrexya 2406 . . . . . . 7 𝑥𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑦 ∈ (𝐹‘(𝑔𝑚)))
11 nfv 1462 . . . . . . . 8 𝑥dom 𝑔 = ∅
12 nffrec.2 . . . . . . . . 9 𝑥𝐴
1312nfcri 2214 . . . . . . . 8 𝑥 𝑦𝐴
1411, 13nfan 1498 . . . . . . 7 𝑥(dom 𝑔 = ∅ ∧ 𝑦𝐴)
1510, 14nfor 1507 . . . . . 6 𝑥(∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑦 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑦𝐴))
1615nfab 2224 . . . . 5 𝑥{𝑦 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑦 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑦𝐴))}
172, 16nfmpt 3878 . . . 4 𝑥(𝑔 ∈ V ↦ {𝑦 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑦 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑦𝐴))})
1817nfrecs 5956 . . 3 𝑥recs((𝑔 ∈ V ↦ {𝑦 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑦 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑦𝐴))}))
1918, 3nfres 4642 . 2 𝑥(recs((𝑔 ∈ V ↦ {𝑦 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑦 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑦𝐴))})) ↾ ω)
201, 19nfcxfr 2217 1 𝑥frec(𝐹, 𝐴)
Colors of variables: wff set class
Syntax hints:  wa 102  wo 662   = wceq 1285  wcel 1434  {cab 2068  wnfc 2207  wrex 2350  Vcvv 2602  c0 3258  cmpt 3847  suc csuc 4128  ωcom 4339  dom cdm 4371  cres 4373  cfv 4932  recscrecs 5953  freccfrec 6039
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-rab 2358  df-v 2604  df-un 2978  df-in 2980  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-br 3794  df-opab 3848  df-mpt 3849  df-xp 4377  df-res 4383  df-iota 4897  df-fv 4940  df-recs 5954  df-frec 6040
This theorem is referenced by:  nfiseq  9528
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