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Theorem nffrec 5921
 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 5918 . 2 frec(𝐹, A) = (recs((g V ↦ {y ∣ (𝑚 𝜔 (dom g = suc 𝑚 y (𝐹‘(g𝑚))) (dom g = ∅ y A))})) ↾ 𝜔)
2 nfcv 2175 . . . . 5 xV
3 nfcv 2175 . . . . . . . 8 x𝜔
4 nfv 1418 . . . . . . . . 9 xdom g = suc 𝑚
5 nffrec.1 . . . . . . . . . . 11 x𝐹
6 nfcv 2175 . . . . . . . . . . 11 x(g𝑚)
75, 6nffv 5128 . . . . . . . . . 10 x(𝐹‘(g𝑚))
87nfcri 2169 . . . . . . . . 9 x y (𝐹‘(g𝑚))
94, 8nfan 1454 . . . . . . . 8 x(dom g = suc 𝑚 y (𝐹‘(g𝑚)))
103, 9nfrexya 2357 . . . . . . 7 x𝑚 𝜔 (dom g = suc 𝑚 y (𝐹‘(g𝑚)))
11 nfv 1418 . . . . . . . 8 xdom g = ∅
12 nffrec.2 . . . . . . . . 9 xA
1312nfcri 2169 . . . . . . . 8 x y A
1411, 13nfan 1454 . . . . . . 7 x(dom g = ∅ y A)
1510, 14nfor 1463 . . . . . 6 x(𝑚 𝜔 (dom g = suc 𝑚 y (𝐹‘(g𝑚))) (dom g = ∅ y A))
1615nfab 2179 . . . . 5 x{y ∣ (𝑚 𝜔 (dom g = suc 𝑚 y (𝐹‘(g𝑚))) (dom g = ∅ y A))}
172, 16nfmpt 3840 . . . 4 x(g V ↦ {y ∣ (𝑚 𝜔 (dom g = suc 𝑚 y (𝐹‘(g𝑚))) (dom g = ∅ y A))})
1817nfrecs 5863 . . 3 xrecs((g V ↦ {y ∣ (𝑚 𝜔 (dom g = suc 𝑚 y (𝐹‘(g𝑚))) (dom g = ∅ y A))}))
1918, 3nfres 4557 . 2 x(recs((g V ↦ {y ∣ (𝑚 𝜔 (dom g = suc 𝑚 y (𝐹‘(g𝑚))) (dom g = ∅ y A))})) ↾ 𝜔)
201, 19nfcxfr 2172 1 xfrec(𝐹, A)
 Colors of variables: wff set class Syntax hints:   ∧ wa 97   ∨ wo 628   = wceq 1242   ∈ wcel 1390  {cab 2023  Ⅎwnfc 2162  ∃wrex 2301  Vcvv 2551  ∅c0 3218   ↦ cmpt 3809  suc csuc 4068  𝜔com 4256  dom cdm 4288   ↾ cres 4290  ‘cfv 4845  recscrecs 5860  freccfrec 5917 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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-un 2916  df-in 2918  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-mpt 3811  df-xp 4294  df-res 4300  df-iota 4810  df-fv 4853  df-recs 5861  df-frec 5918 This theorem is referenced by:  nfiseq  8878
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