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Theorem nffrec 6399
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 6394 . 2 frec(𝐹, 𝐴) = (recs((𝑔 ∈ V ↦ {𝑦 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑦 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑦𝐴))})) ↾ ω)
2 nfcv 2319 . . . . 5 𝑥V
3 nfcv 2319 . . . . . . . 8 𝑥ω
4 nfv 1528 . . . . . . . . 9 𝑥dom 𝑔 = suc 𝑚
5 nffrec.1 . . . . . . . . . . 11 𝑥𝐹
6 nfcv 2319 . . . . . . . . . . 11 𝑥(𝑔𝑚)
75, 6nffv 5527 . . . . . . . . . 10 𝑥(𝐹‘(𝑔𝑚))
87nfcri 2313 . . . . . . . . 9 𝑥 𝑦 ∈ (𝐹‘(𝑔𝑚))
94, 8nfan 1565 . . . . . . . 8 𝑥(dom 𝑔 = suc 𝑚𝑦 ∈ (𝐹‘(𝑔𝑚)))
103, 9nfrexya 2518 . . . . . . 7 𝑥𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑦 ∈ (𝐹‘(𝑔𝑚)))
11 nfv 1528 . . . . . . . 8 𝑥dom 𝑔 = ∅
12 nffrec.2 . . . . . . . . 9 𝑥𝐴
1312nfcri 2313 . . . . . . . 8 𝑥 𝑦𝐴
1411, 13nfan 1565 . . . . . . 7 𝑥(dom 𝑔 = ∅ ∧ 𝑦𝐴)
1510, 14nfor 1574 . . . . . 6 𝑥(∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑦 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑦𝐴))
1615nfab 2324 . . . . 5 𝑥{𝑦 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑦 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑦𝐴))}
172, 16nfmpt 4097 . . . 4 𝑥(𝑔 ∈ V ↦ {𝑦 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑦 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑦𝐴))})
1817nfrecs 6310 . . 3 𝑥recs((𝑔 ∈ V ↦ {𝑦 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑦 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑦𝐴))}))
1918, 3nfres 4911 . 2 𝑥(recs((𝑔 ∈ V ↦ {𝑦 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑦 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑦𝐴))})) ↾ ω)
201, 19nfcxfr 2316 1 𝑥frec(𝐹, 𝐴)
Colors of variables: wff set class
Syntax hints:  wa 104  wo 708   = wceq 1353  wcel 2148  {cab 2163  wnfc 2306  wrex 2456  Vcvv 2739  c0 3424  cmpt 4066  suc csuc 4367  ωcom 4591  dom cdm 4628  cres 4630  cfv 5218  recscrecs 6307  freccfrec 6393
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2741  df-un 3135  df-in 3137  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-mpt 4068  df-xp 4634  df-res 4640  df-iota 5180  df-fv 5226  df-recs 6308  df-frec 6394
This theorem is referenced by:  nfseq  10457
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