Theorem nffrec 6355

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 6350	. 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 2306	. . . . 5 ⊢ Ⅎ𝑥V
3		nfcv 2306	. . . . . . . 8 ⊢ Ⅎ𝑥ω
4		nfv 1515	. . . . . . . . 9 ⊢ Ⅎ𝑥dom 𝑔 = suc 𝑚
5		nffrec.1	. . . . . . . . . . 11 ⊢ Ⅎ𝑥𝐹
6		nfcv 2306	. . . . . . . . . . 11 ⊢ Ⅎ𝑥(𝑔‘𝑚)
7	5, 6	nffv 5490	. . . . . . . . . 10 ⊢ Ⅎ𝑥(𝐹‘(𝑔‘𝑚))
8	7	nfcri 2300	. . . . . . . . 9 ⊢ Ⅎ𝑥 𝑦 ∈ (𝐹‘(𝑔‘𝑚))
9	4, 8	nfan 1552	. . . . . . . 8 ⊢ Ⅎ𝑥(dom 𝑔 = suc 𝑚 ∧ 𝑦 ∈ (𝐹‘(𝑔‘𝑚)))
10	3, 9	nfrexya 2505	. . . . . . 7 ⊢ Ⅎ𝑥∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑦 ∈ (𝐹‘(𝑔‘𝑚)))
11		nfv 1515	. . . . . . . 8 ⊢ Ⅎ𝑥dom 𝑔 = ∅
12		nffrec.2	. . . . . . . . 9 ⊢ Ⅎ𝑥𝐴
13	12	nfcri 2300	. . . . . . . 8 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴
14	11, 13	nfan 1552	. . . . . . 7 ⊢ Ⅎ𝑥(dom 𝑔 = ∅ ∧ 𝑦 ∈ 𝐴)
15	10, 14	nfor 1561	. . . . . 6 ⊢ Ⅎ𝑥(∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑦 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑦 ∈ 𝐴))
16	15	nfab 2311	. . . . 5 ⊢ Ⅎ𝑥{𝑦 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑦 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑦 ∈ 𝐴))}
17	2, 16	nfmpt 4068	. . . 4 ⊢ Ⅎ𝑥(𝑔 ∈ V ↦ {𝑦 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑦 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑦 ∈ 𝐴))})
18	17	nfrecs 6266	. . 3 ⊢ Ⅎ𝑥recs((𝑔 ∈ V ↦ {𝑦 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑦 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑦 ∈ 𝐴))}))
19	18, 3	nfres 4880	. 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 2303	1 ⊢ Ⅎ𝑥frec(𝐹, 𝐴)

Syntax hints:   ∧ wa 103   ∨ wo 698   = wceq 1342   ∈ wcel 2135  {cab 2150  Ⅎwnfc 2293  ∃wrex 2443  Vcvv 2721  ∅c0 3404   ↦ cmpt 4037  suc csuc 4337  ωcom 4561  dom cdm 4598   ↾ cres 4600  ‘cfv 5182  recscrecs 6263  freccfrec 6349

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-ext 2146

This theorem depends on definitions:  df-bi 116  df-3an 969  df-tru 1345  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ral 2447  df-rex 2448  df-rab 2451  df-v 2723  df-un 3115  df-in 3117  df-sn 3576  df-pr 3577  df-op 3579  df-uni 3784  df-br 3977  df-opab 4038  df-mpt 4039  df-xp 4604  df-res 4610  df-iota 5147  df-fv 5190  df-recs 6264  df-frec 6350

This theorem is referenced by:  nfseq  10380