Theorem nffrec 6375

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 6370	. 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 2312	. . . . 5 ⊢ Ⅎ𝑥V
3		nfcv 2312	. . . . . . . 8 ⊢ Ⅎ𝑥ω
4		nfv 1521	. . . . . . . . 9 ⊢ Ⅎ𝑥dom 𝑔 = suc 𝑚
5		nffrec.1	. . . . . . . . . . 11 ⊢ Ⅎ𝑥𝐹
6		nfcv 2312	. . . . . . . . . . 11 ⊢ Ⅎ𝑥(𝑔‘𝑚)
7	5, 6	nffv 5506	. . . . . . . . . 10 ⊢ Ⅎ𝑥(𝐹‘(𝑔‘𝑚))
8	7	nfcri 2306	. . . . . . . . 9 ⊢ Ⅎ𝑥 𝑦 ∈ (𝐹‘(𝑔‘𝑚))
9	4, 8	nfan 1558	. . . . . . . 8 ⊢ Ⅎ𝑥(dom 𝑔 = suc 𝑚 ∧ 𝑦 ∈ (𝐹‘(𝑔‘𝑚)))
10	3, 9	nfrexya 2511	. . . . . . 7 ⊢ Ⅎ𝑥∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑦 ∈ (𝐹‘(𝑔‘𝑚)))
11		nfv 1521	. . . . . . . 8 ⊢ Ⅎ𝑥dom 𝑔 = ∅
12		nffrec.2	. . . . . . . . 9 ⊢ Ⅎ𝑥𝐴
13	12	nfcri 2306	. . . . . . . 8 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴
14	11, 13	nfan 1558	. . . . . . 7 ⊢ Ⅎ𝑥(dom 𝑔 = ∅ ∧ 𝑦 ∈ 𝐴)
15	10, 14	nfor 1567	. . . . . 6 ⊢ Ⅎ𝑥(∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑦 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑦 ∈ 𝐴))
16	15	nfab 2317	. . . . 5 ⊢ Ⅎ𝑥{𝑦 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑦 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑦 ∈ 𝐴))}
17	2, 16	nfmpt 4081	. . . 4 ⊢ Ⅎ𝑥(𝑔 ∈ V ↦ {𝑦 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑦 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑦 ∈ 𝐴))})
18	17	nfrecs 6286	. . 3 ⊢ Ⅎ𝑥recs((𝑔 ∈ V ↦ {𝑦 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑦 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑦 ∈ 𝐴))}))
19	18, 3	nfres 4893	. 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 2309	1 ⊢ Ⅎ𝑥frec(𝐹, 𝐴)

Syntax hints:   ∧ wa 103   ∨ wo 703   = wceq 1348   ∈ wcel 2141  {cab 2156  Ⅎwnfc 2299  ∃wrex 2449  Vcvv 2730  ∅c0 3414   ↦ cmpt 4050  suc csuc 4350  ωcom 4574  dom cdm 4611   ↾ cres 4613  ‘cfv 5198  recscrecs 6283  freccfrec 6369

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-un 3125  df-in 3127  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-mpt 4052  df-xp 4617  df-res 4623  df-iota 5160  df-fv 5206  df-recs 6284  df-frec 6370

This theorem is referenced by:  nfseq  10411