Theorem nffrec 6605

Description: Bound-variable hypothesis builder for the finite recursive definition generator. (Contributed by Jim Kingdon, 30-May-2020.)

Hypotheses

Ref	Expression
nffrec.1	⊢ Ⅎ𝑥𝐹
nffrec.2	⊢ Ⅎ𝑥𝐴

Assertion

Ref	Expression
nffrec	⊢ Ⅎ𝑥frec(𝐹, 𝐴)

Proof of Theorem nffrec

Dummy variables 𝑔 𝑚 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-frec 6600	. 2 ⊢ frec(𝐹, 𝐴) = (recs((𝑔 ∈ V ↦ {𝑦 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑦 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑦 ∈ 𝐴))})) ↾ ω)
2		nfcv 2375	. . . . 5 ⊢ Ⅎ𝑥V
3		nfcv 2375	. . . . . . . 8 ⊢ Ⅎ𝑥ω
4		nfv 1577	. . . . . . . . 9 ⊢ Ⅎ𝑥dom 𝑔 = suc 𝑚
5		nffrec.1	. . . . . . . . . . 11 ⊢ Ⅎ𝑥𝐹
6		nfcv 2375	. . . . . . . . . . 11 ⊢ Ⅎ𝑥(𝑔‘𝑚)
7	5, 6	nffv 5658	. . . . . . . . . 10 ⊢ Ⅎ𝑥(𝐹‘(𝑔‘𝑚))
8	7	nfcri 2369	. . . . . . . . 9 ⊢ Ⅎ𝑥 𝑦 ∈ (𝐹‘(𝑔‘𝑚))
9	4, 8	nfan 1614	. . . . . . . 8 ⊢ Ⅎ𝑥(dom 𝑔 = suc 𝑚 ∧ 𝑦 ∈ (𝐹‘(𝑔‘𝑚)))
10	3, 9	nfrexya 2574	. . . . . . 7 ⊢ Ⅎ𝑥∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑦 ∈ (𝐹‘(𝑔‘𝑚)))
11		nfv 1577	. . . . . . . 8 ⊢ Ⅎ𝑥dom 𝑔 = ∅
12		nffrec.2	. . . . . . . . 9 ⊢ Ⅎ𝑥𝐴
13	12	nfcri 2369	. . . . . . . 8 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴
14	11, 13	nfan 1614	. . . . . . 7 ⊢ Ⅎ𝑥(dom 𝑔 = ∅ ∧ 𝑦 ∈ 𝐴)
15	10, 14	nfor 1623	. . . . . 6 ⊢ Ⅎ𝑥(∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑦 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑦 ∈ 𝐴))
16	15	nfab 2380	. . . . 5 ⊢ Ⅎ𝑥{𝑦 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑦 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑦 ∈ 𝐴))}
17	2, 16	nfmpt 4186	. . . 4 ⊢ Ⅎ𝑥(𝑔 ∈ V ↦ {𝑦 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑦 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑦 ∈ 𝐴))})
18	17	nfrecs 6516	. . 3 ⊢ Ⅎ𝑥recs((𝑔 ∈ V ↦ {𝑦 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑦 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑦 ∈ 𝐴))}))
19	18, 3	nfres 5021	. 2 ⊢ Ⅎ𝑥(recs((𝑔 ∈ V ↦ {𝑦 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑦 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑦 ∈ 𝐴))})) ↾ ω)
20	1, 19	nfcxfr 2372	1 ⊢ Ⅎ𝑥frec(𝐹, 𝐴)

Syntax hints:   ∧ wa 104   ∨ wo 716   = wceq 1398   ∈ wcel 2202  {cab 2217  Ⅎwnfc 2362  ∃wrex 2512  Vcvv 2803  ∅c0 3496   ↦ cmpt 4155  suc csuc 4468  ωcom 4694  dom cdm 4731   ↾ cres 4733  ‘cfv 5333  recscrecs 6513  freccfrec 6599

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-rab 2520  df-v 2805  df-un 3205  df-in 3207  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-mpt 4157  df-xp 4737  df-res 4743  df-iota 5293  df-fv 5341  df-recs 6514  df-frec 6600

This theorem is referenced by:  nfseq  10765