Theorem nffrec 5921

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 5918	. 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 2175	. . . . 5 ⊢ ℲxV
3		nfcv 2175	. . . . . . . 8 ⊢ Ⅎx𝜔
4		nfv 1418	. . . . . . . . 9 ⊢ Ⅎxdom g = suc 𝑚
5		nffrec.1	. . . . . . . . . . 11 ⊢ Ⅎx𝐹
6		nfcv 2175	. . . . . . . . . . 11 ⊢ Ⅎx(g‘𝑚)
7	5, 6	nffv 5128	. . . . . . . . . 10 ⊢ Ⅎx(𝐹‘(g‘𝑚))
8	7	nfcri 2169	. . . . . . . . 9 ⊢ Ⅎx y ∈ (𝐹‘(g‘𝑚))
9	4, 8	nfan 1454	. . . . . . . 8 ⊢ Ⅎx(dom g = suc 𝑚 ∧ y ∈ (𝐹‘(g‘𝑚)))
10	3, 9	nfrexya 2357	. . . . . . 7 ⊢ Ⅎx∃𝑚 ∈ 𝜔 (dom g = suc 𝑚 ∧ y ∈ (𝐹‘(g‘𝑚)))
11		nfv 1418	. . . . . . . 8 ⊢ Ⅎxdom g = ∅
12		nffrec.2	. . . . . . . . 9 ⊢ ℲxA
13	12	nfcri 2169	. . . . . . . 8 ⊢ Ⅎx y ∈ A
14	11, 13	nfan 1454	. . . . . . 7 ⊢ Ⅎx(dom g = ∅ ∧ y ∈ A)
15	10, 14	nfor 1463	. . . . . 6 ⊢ Ⅎx(∃𝑚 ∈ 𝜔 (dom g = suc 𝑚 ∧ y ∈ (𝐹‘(g‘𝑚))) ∨ (dom g = ∅ ∧ y ∈ A))
16	15	nfab 2179	. . . . 5 ⊢ Ⅎx{y ∣ (∃𝑚 ∈ 𝜔 (dom g = suc 𝑚 ∧ y ∈ (𝐹‘(g‘𝑚))) ∨ (dom g = ∅ ∧ y ∈ A))}
17	2, 16	nfmpt 3840	. . . 4 ⊢ Ⅎx(g ∈ V ↦ {y ∣ (∃𝑚 ∈ 𝜔 (dom g = suc 𝑚 ∧ y ∈ (𝐹‘(g‘𝑚))) ∨ (dom g = ∅ ∧ y ∈ A))})
18	17	nfrecs 5863	. . 3 ⊢ Ⅎxrecs((g ∈ V ↦ {y ∣ (∃𝑚 ∈ 𝜔 (dom g = suc 𝑚 ∧ y ∈ (𝐹‘(g‘𝑚))) ∨ (dom g = ∅ ∧ y ∈ A))}))
19	18, 3	nfres 4557	. 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 2172	1 ⊢ Ⅎxfrec(𝐹, A)

Syntax hints:   ∧ wa 97   ∨ wo 628   = wceq 1242   ∈ wcel 1390  {cab 2023  Ⅎwnfc 2162  ∃wrex 2301  Vcvv 2551  ∅c0 3218   ↦ cmpt 3809  suc csuc 4068  𝜔com 4256  dom cdm 4288   ↾ cres 4290  ‘cfv 4845  recscrecs 5860  freccfrec 5917

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-un 2916  df-in 2918  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-mpt 3811  df-xp 4294  df-res 4300  df-iota 4810  df-fv 4853  df-recs 5861  df-frec 5918

This theorem is referenced by:  nfiseq  8898