Theorem nfrecs 6423

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 6421	. 2 ⊢ recs(𝐹) = ∪ {𝑎 ∣ ∃𝑏 ∈ On (𝑎 Fn 𝑏 ∧ ∀𝑐 ∈ 𝑏 (𝑎‘𝑐) = (𝐹‘(𝑎 ↾ 𝑐)))}
2		nfcv 2352	. . . . 5 ⊢ Ⅎ𝑥On
3		nfv 1554	. . . . . 6 ⊢ Ⅎ𝑥 𝑎 Fn 𝑏
4		nfcv 2352	. . . . . . 7 ⊢ Ⅎ𝑥𝑏
5		nfrecs.f	. . . . . . . . 9 ⊢ Ⅎ𝑥𝐹
6		nfcv 2352	. . . . . . . . 9 ⊢ Ⅎ𝑥(𝑎 ↾ 𝑐)
7	5, 6	nffv 5613	. . . . . . . 8 ⊢ Ⅎ𝑥(𝐹‘(𝑎 ↾ 𝑐))
8	7	nfeq2 2364	. . . . . . 7 ⊢ Ⅎ𝑥(𝑎‘𝑐) = (𝐹‘(𝑎 ↾ 𝑐))
9	4, 8	nfralxy 2548	. . . . . 6 ⊢ Ⅎ𝑥∀𝑐 ∈ 𝑏 (𝑎‘𝑐) = (𝐹‘(𝑎 ↾ 𝑐))
10	3, 9	nfan 1591	. . . . 5 ⊢ Ⅎ𝑥(𝑎 Fn 𝑏 ∧ ∀𝑐 ∈ 𝑏 (𝑎‘𝑐) = (𝐹‘(𝑎 ↾ 𝑐)))
11	2, 10	nfrexw 2549	. . . 4 ⊢ Ⅎ𝑥∃𝑏 ∈ On (𝑎 Fn 𝑏 ∧ ∀𝑐 ∈ 𝑏 (𝑎‘𝑐) = (𝐹‘(𝑎 ↾ 𝑐)))
12	11	nfab 2357	. . 3 ⊢ Ⅎ𝑥{𝑎 ∣ ∃𝑏 ∈ On (𝑎 Fn 𝑏 ∧ ∀𝑐 ∈ 𝑏 (𝑎‘𝑐) = (𝐹‘(𝑎 ↾ 𝑐)))}
13	12	nfuni 3873	. 2 ⊢ Ⅎ𝑥∪ {𝑎 ∣ ∃𝑏 ∈ On (𝑎 Fn 𝑏 ∧ ∀𝑐 ∈ 𝑏 (𝑎‘𝑐) = (𝐹‘(𝑎 ↾ 𝑐)))}
14	1, 13	nfcxfr 2349	1 ⊢ Ⅎ𝑥recs(𝐹)

Syntax hints:   ∧ wa 104   = wceq 1375  {cab 2195  Ⅎwnfc 2339  ∀wral 2488  ∃wrex 2489  ∪ cuni 3867  Oncon0 4431   ↾ cres 4698   Fn wfn 5289  ‘cfv 5294  recscrecs 6420

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 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-ext 2191

This theorem depends on definitions:  df-bi 117  df-3an 985  df-tru 1378  df-nf 1487  df-sb 1789  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-ral 2493  df-rex 2494  df-v 2781  df-un 3181  df-sn 3652  df-pr 3653  df-op 3655  df-uni 3868  df-br 4063  df-iota 5254  df-fv 5302  df-recs 6421

This theorem is referenced by:  nffrec  6512