Theorem nffrec 6005

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 6001	. 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 2219	. . . . 5 ⊢ Ⅎ𝑥V
3		nfcv 2219	. . . . . . . 8 ⊢ Ⅎ𝑥ω
4		nfv 1461	. . . . . . . . 9 ⊢ Ⅎ𝑥dom 𝑔 = suc 𝑚
5		nffrec.1	. . . . . . . . . . 11 ⊢ Ⅎ𝑥𝐹
6		nfcv 2219	. . . . . . . . . . 11 ⊢ Ⅎ𝑥(𝑔‘𝑚)
7	5, 6	nffv 5205	. . . . . . . . . 10 ⊢ Ⅎ𝑥(𝐹‘(𝑔‘𝑚))
8	7	nfcri 2213	. . . . . . . . 9 ⊢ Ⅎ𝑥 𝑦 ∈ (𝐹‘(𝑔‘𝑚))
9	4, 8	nfan 1497	. . . . . . . 8 ⊢ Ⅎ𝑥(dom 𝑔 = suc 𝑚 ∧ 𝑦 ∈ (𝐹‘(𝑔‘𝑚)))
10	3, 9	nfrexya 2405	. . . . . . 7 ⊢ Ⅎ𝑥∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑦 ∈ (𝐹‘(𝑔‘𝑚)))
11		nfv 1461	. . . . . . . 8 ⊢ Ⅎ𝑥dom 𝑔 = ∅
12		nffrec.2	. . . . . . . . 9 ⊢ Ⅎ𝑥𝐴
13	12	nfcri 2213	. . . . . . . 8 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴
14	11, 13	nfan 1497	. . . . . . 7 ⊢ Ⅎ𝑥(dom 𝑔 = ∅ ∧ 𝑦 ∈ 𝐴)
15	10, 14	nfor 1506	. . . . . 6 ⊢ Ⅎ𝑥(∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑦 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑦 ∈ 𝐴))
16	15	nfab 2223	. . . . 5 ⊢ Ⅎ𝑥{𝑦 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑦 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑦 ∈ 𝐴))}
17	2, 16	nfmpt 3870	. . . 4 ⊢ Ⅎ𝑥(𝑔 ∈ V ↦ {𝑦 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑦 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑦 ∈ 𝐴))})
18	17	nfrecs 5945	. . 3 ⊢ Ⅎ𝑥recs((𝑔 ∈ V ↦ {𝑦 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑦 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑦 ∈ 𝐴))}))
19	18, 3	nfres 4632	. 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 2216	1 ⊢ Ⅎ𝑥frec(𝐹, 𝐴)

Syntax hints:   ∧ wa 102   ∨ wo 661   = wceq 1284   ∈ wcel 1433  {cab 2067  Ⅎwnfc 2206  ∃wrex 2349  Vcvv 2601  ∅c0 3251   ↦ cmpt 3839  suc csuc 4120  ωcom 4331  dom cdm 4363   ↾ cres 4365  ‘cfv 4922  recscrecs 5942  freccfrec 6000

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-rab 2357  df-v 2603  df-un 2977  df-in 2979  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-mpt 3841  df-xp 4369  df-res 4375  df-iota 4887  df-fv 4930  df-recs 5943  df-frec 6001

This theorem is referenced by:  nfiseq  9438