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Theorem nffrec 5982
 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 5978 . 2 frec(𝐹, 𝐴) = (recs((𝑔 ∈ V ↦ {𝑦 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑦 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑦𝐴))})) ↾ ω)
2 nfcv 2178 . . . . 5 𝑥V
3 nfcv 2178 . . . . . . . 8 𝑥ω
4 nfv 1421 . . . . . . . . 9 𝑥dom 𝑔 = suc 𝑚
5 nffrec.1 . . . . . . . . . . 11 𝑥𝐹
6 nfcv 2178 . . . . . . . . . . 11 𝑥(𝑔𝑚)
75, 6nffv 5185 . . . . . . . . . 10 𝑥(𝐹‘(𝑔𝑚))
87nfcri 2172 . . . . . . . . 9 𝑥 𝑦 ∈ (𝐹‘(𝑔𝑚))
94, 8nfan 1457 . . . . . . . 8 𝑥(dom 𝑔 = suc 𝑚𝑦 ∈ (𝐹‘(𝑔𝑚)))
103, 9nfrexya 2363 . . . . . . 7 𝑥𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑦 ∈ (𝐹‘(𝑔𝑚)))
11 nfv 1421 . . . . . . . 8 𝑥dom 𝑔 = ∅
12 nffrec.2 . . . . . . . . 9 𝑥𝐴
1312nfcri 2172 . . . . . . . 8 𝑥 𝑦𝐴
1411, 13nfan 1457 . . . . . . 7 𝑥(dom 𝑔 = ∅ ∧ 𝑦𝐴)
1510, 14nfor 1466 . . . . . 6 𝑥(∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑦 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑦𝐴))
1615nfab 2182 . . . . 5 𝑥{𝑦 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑦 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑦𝐴))}
172, 16nfmpt 3849 . . . 4 𝑥(𝑔 ∈ V ↦ {𝑦 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑦 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑦𝐴))})
1817nfrecs 5922 . . 3 𝑥recs((𝑔 ∈ V ↦ {𝑦 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑦 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑦𝐴))}))
1918, 3nfres 4614 . 2 𝑥(recs((𝑔 ∈ V ↦ {𝑦 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑦 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑦𝐴))})) ↾ ω)
201, 19nfcxfr 2175 1 𝑥frec(𝐹, 𝐴)
 Colors of variables: wff set class Syntax hints:   ∧ wa 97   ∨ wo 629   = wceq 1243   ∈ wcel 1393  {cab 2026  Ⅎwnfc 2165  ∃wrex 2307  Vcvv 2557  ∅c0 3224   ↦ cmpt 3818  suc csuc 4102  ωcom 4313  dom cdm 4345   ↾ cres 4347  ‘cfv 4902  recscrecs 5919  freccfrec 5977 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-rab 2315  df-v 2559  df-un 2922  df-in 2924  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-mpt 3820  df-xp 4351  df-res 4357  df-iota 4867  df-fv 4910  df-recs 5920  df-frec 5978 This theorem is referenced by:  nfiseq  9218
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