Theorem nffrec 6454

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 6449	. 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 2339	. . . . 5 ⊢ Ⅎ𝑥V
3		nfcv 2339	. . . . . . . 8 ⊢ Ⅎ𝑥ω
4		nfv 1542	. . . . . . . . 9 ⊢ Ⅎ𝑥dom 𝑔 = suc 𝑚
5		nffrec.1	. . . . . . . . . . 11 ⊢ Ⅎ𝑥𝐹
6		nfcv 2339	. . . . . . . . . . 11 ⊢ Ⅎ𝑥(𝑔‘𝑚)
7	5, 6	nffv 5568	. . . . . . . . . 10 ⊢ Ⅎ𝑥(𝐹‘(𝑔‘𝑚))
8	7	nfcri 2333	. . . . . . . . 9 ⊢ Ⅎ𝑥 𝑦 ∈ (𝐹‘(𝑔‘𝑚))
9	4, 8	nfan 1579	. . . . . . . 8 ⊢ Ⅎ𝑥(dom 𝑔 = suc 𝑚 ∧ 𝑦 ∈ (𝐹‘(𝑔‘𝑚)))
10	3, 9	nfrexya 2538	. . . . . . 7 ⊢ Ⅎ𝑥∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑦 ∈ (𝐹‘(𝑔‘𝑚)))
11		nfv 1542	. . . . . . . 8 ⊢ Ⅎ𝑥dom 𝑔 = ∅
12		nffrec.2	. . . . . . . . 9 ⊢ Ⅎ𝑥𝐴
13	12	nfcri 2333	. . . . . . . 8 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴
14	11, 13	nfan 1579	. . . . . . 7 ⊢ Ⅎ𝑥(dom 𝑔 = ∅ ∧ 𝑦 ∈ 𝐴)
15	10, 14	nfor 1588	. . . . . 6 ⊢ Ⅎ𝑥(∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑦 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑦 ∈ 𝐴))
16	15	nfab 2344	. . . . 5 ⊢ Ⅎ𝑥{𝑦 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑦 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑦 ∈ 𝐴))}
17	2, 16	nfmpt 4125	. . . 4 ⊢ Ⅎ𝑥(𝑔 ∈ V ↦ {𝑦 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑦 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑦 ∈ 𝐴))})
18	17	nfrecs 6365	. . 3 ⊢ Ⅎ𝑥recs((𝑔 ∈ V ↦ {𝑦 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑦 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑦 ∈ 𝐴))}))
19	18, 3	nfres 4948	. 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 2336	1 ⊢ Ⅎ𝑥frec(𝐹, 𝐴)

Syntax hints:   ∧ wa 104   ∨ wo 709   = wceq 1364   ∈ wcel 2167  {cab 2182  Ⅎwnfc 2326  ∃wrex 2476  Vcvv 2763  ∅c0 3450   ↦ cmpt 4094  suc csuc 4400  ωcom 4626  dom cdm 4663   ↾ cres 4665  ‘cfv 5258  recscrecs 6362  freccfrec 6448

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-un 3161  df-in 3163  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-opab 4095  df-mpt 4096  df-xp 4669  df-res 4675  df-iota 5219  df-fv 5266  df-recs 6363  df-frec 6449

This theorem is referenced by:  nfseq  10549