Theorem nfrecs 6332

Description: Bound-variable hypothesis builder for recs. (Contributed by Stefan O'Rear, 18-Jan-2015.)

Hypothesis

Ref	Expression
nfrecs.f	⊢ Ⅎ𝑥𝐹

Assertion

Ref	Expression
nfrecs	⊢ Ⅎ𝑥recs(𝐹)

Proof of Theorem nfrecs

Dummy variables 𝑎 𝑏 𝑐 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-recs 6330	. 2 ⊢ recs(𝐹) = ∪ {𝑎 ∣ ∃𝑏 ∈ On (𝑎 Fn 𝑏 ∧ ∀𝑐 ∈ 𝑏 (𝑎‘𝑐) = (𝐹‘(𝑎 ↾ 𝑐)))}
2		nfcv 2332	. . . . 5 ⊢ Ⅎ𝑥On
3		nfv 1539	. . . . . 6 ⊢ Ⅎ𝑥 𝑎 Fn 𝑏
4		nfcv 2332	. . . . . . 7 ⊢ Ⅎ𝑥𝑏
5		nfrecs.f	. . . . . . . . 9 ⊢ Ⅎ𝑥𝐹
6		nfcv 2332	. . . . . . . . 9 ⊢ Ⅎ𝑥(𝑎 ↾ 𝑐)
7	5, 6	nffv 5544	. . . . . . . 8 ⊢ Ⅎ𝑥(𝐹‘(𝑎 ↾ 𝑐))
8	7	nfeq2 2344	. . . . . . 7 ⊢ Ⅎ𝑥(𝑎‘𝑐) = (𝐹‘(𝑎 ↾ 𝑐))
9	4, 8	nfralxy 2528	. . . . . 6 ⊢ Ⅎ𝑥∀𝑐 ∈ 𝑏 (𝑎‘𝑐) = (𝐹‘(𝑎 ↾ 𝑐))
10	3, 9	nfan 1576	. . . . 5 ⊢ Ⅎ𝑥(𝑎 Fn 𝑏 ∧ ∀𝑐 ∈ 𝑏 (𝑎‘𝑐) = (𝐹‘(𝑎 ↾ 𝑐)))
11	2, 10	nfrexxy 2529	. . . 4 ⊢ Ⅎ𝑥∃𝑏 ∈ On (𝑎 Fn 𝑏 ∧ ∀𝑐 ∈ 𝑏 (𝑎‘𝑐) = (𝐹‘(𝑎 ↾ 𝑐)))
12	11	nfab 2337	. . 3 ⊢ Ⅎ𝑥{𝑎 ∣ ∃𝑏 ∈ On (𝑎 Fn 𝑏 ∧ ∀𝑐 ∈ 𝑏 (𝑎‘𝑐) = (𝐹‘(𝑎 ↾ 𝑐)))}
13	12	nfuni 3830	. 2 ⊢ Ⅎ𝑥∪ {𝑎 ∣ ∃𝑏 ∈ On (𝑎 Fn 𝑏 ∧ ∀𝑐 ∈ 𝑏 (𝑎‘𝑐) = (𝐹‘(𝑎 ↾ 𝑐)))}
14	1, 13	nfcxfr 2329	1 ⊢ Ⅎ𝑥recs(𝐹)

Syntax hints:   ∧ wa 104   = wceq 1364  {cab 2175  Ⅎwnfc 2319  ∀wral 2468  ∃wrex 2469  ∪ cuni 3824  Oncon0 4381   ↾ cres 4646   Fn wfn 5230  ‘cfv 5235  recscrecs 6329

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-un 3148  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-iota 5196  df-fv 5243  df-recs 6330

This theorem is referenced by:  nffrec  6421