Theorem nfrecs 6374

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 6372	. 2 ⊢ recs(𝐹) = ∪ {𝑎 ∣ ∃𝑏 ∈ On (𝑎 Fn 𝑏 ∧ ∀𝑐 ∈ 𝑏 (𝑎‘𝑐) = (𝐹‘(𝑎 ↾ 𝑐)))}
2		nfcv 2339	. . . . 5 ⊢ Ⅎ𝑥On
3		nfv 1542	. . . . . 6 ⊢ Ⅎ𝑥 𝑎 Fn 𝑏
4		nfcv 2339	. . . . . . 7 ⊢ Ⅎ𝑥𝑏
5		nfrecs.f	. . . . . . . . 9 ⊢ Ⅎ𝑥𝐹
6		nfcv 2339	. . . . . . . . 9 ⊢ Ⅎ𝑥(𝑎 ↾ 𝑐)
7	5, 6	nffv 5571	. . . . . . . 8 ⊢ Ⅎ𝑥(𝐹‘(𝑎 ↾ 𝑐))
8	7	nfeq2 2351	. . . . . . 7 ⊢ Ⅎ𝑥(𝑎‘𝑐) = (𝐹‘(𝑎 ↾ 𝑐))
9	4, 8	nfralxy 2535	. . . . . 6 ⊢ Ⅎ𝑥∀𝑐 ∈ 𝑏 (𝑎‘𝑐) = (𝐹‘(𝑎 ↾ 𝑐))
10	3, 9	nfan 1579	. . . . 5 ⊢ Ⅎ𝑥(𝑎 Fn 𝑏 ∧ ∀𝑐 ∈ 𝑏 (𝑎‘𝑐) = (𝐹‘(𝑎 ↾ 𝑐)))
11	2, 10	nfrexw 2536	. . . 4 ⊢ Ⅎ𝑥∃𝑏 ∈ On (𝑎 Fn 𝑏 ∧ ∀𝑐 ∈ 𝑏 (𝑎‘𝑐) = (𝐹‘(𝑎 ↾ 𝑐)))
12	11	nfab 2344	. . 3 ⊢ Ⅎ𝑥{𝑎 ∣ ∃𝑏 ∈ On (𝑎 Fn 𝑏 ∧ ∀𝑐 ∈ 𝑏 (𝑎‘𝑐) = (𝐹‘(𝑎 ↾ 𝑐)))}
13	12	nfuni 3846	. 2 ⊢ Ⅎ𝑥∪ {𝑎 ∣ ∃𝑏 ∈ On (𝑎 Fn 𝑏 ∧ ∀𝑐 ∈ 𝑏 (𝑎‘𝑐) = (𝐹‘(𝑎 ↾ 𝑐)))}
14	1, 13	nfcxfr 2336	1 ⊢ Ⅎ𝑥recs(𝐹)

Syntax hints:   ∧ wa 104   = wceq 1364  {cab 2182  Ⅎwnfc 2326  ∀wral 2475  ∃wrex 2476  ∪ cuni 3840  Oncon0 4399   ↾ cres 4666   Fn wfn 5254  ‘cfv 5259  recscrecs 6371

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-v 2765  df-un 3161  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-br 4035  df-iota 5220  df-fv 5267  df-recs 6372

This theorem is referenced by:  nffrec  6463