Theorem nfrecs 6158

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 6156	. 2 ⊢ recs(𝐹) = ∪ {𝑎 ∣ ∃𝑏 ∈ On (𝑎 Fn 𝑏 ∧ ∀𝑐 ∈ 𝑏 (𝑎‘𝑐) = (𝐹‘(𝑎 ↾ 𝑐)))}
2		nfcv 2255	. . . . 5 ⊢ Ⅎ𝑥On
3		nfv 1491	. . . . . 6 ⊢ Ⅎ𝑥 𝑎 Fn 𝑏
4		nfcv 2255	. . . . . . 7 ⊢ Ⅎ𝑥𝑏
5		nfrecs.f	. . . . . . . . 9 ⊢ Ⅎ𝑥𝐹
6		nfcv 2255	. . . . . . . . 9 ⊢ Ⅎ𝑥(𝑎 ↾ 𝑐)
7	5, 6	nffv 5385	. . . . . . . 8 ⊢ Ⅎ𝑥(𝐹‘(𝑎 ↾ 𝑐))
8	7	nfeq2 2267	. . . . . . 7 ⊢ Ⅎ𝑥(𝑎‘𝑐) = (𝐹‘(𝑎 ↾ 𝑐))
9	4, 8	nfralxy 2445	. . . . . 6 ⊢ Ⅎ𝑥∀𝑐 ∈ 𝑏 (𝑎‘𝑐) = (𝐹‘(𝑎 ↾ 𝑐))
10	3, 9	nfan 1527	. . . . 5 ⊢ Ⅎ𝑥(𝑎 Fn 𝑏 ∧ ∀𝑐 ∈ 𝑏 (𝑎‘𝑐) = (𝐹‘(𝑎 ↾ 𝑐)))
11	2, 10	nfrexxy 2446	. . . 4 ⊢ Ⅎ𝑥∃𝑏 ∈ On (𝑎 Fn 𝑏 ∧ ∀𝑐 ∈ 𝑏 (𝑎‘𝑐) = (𝐹‘(𝑎 ↾ 𝑐)))
12	11	nfab 2260	. . 3 ⊢ Ⅎ𝑥{𝑎 ∣ ∃𝑏 ∈ On (𝑎 Fn 𝑏 ∧ ∀𝑐 ∈ 𝑏 (𝑎‘𝑐) = (𝐹‘(𝑎 ↾ 𝑐)))}
13	12	nfuni 3708	. 2 ⊢ Ⅎ𝑥∪ {𝑎 ∣ ∃𝑏 ∈ On (𝑎 Fn 𝑏 ∧ ∀𝑐 ∈ 𝑏 (𝑎‘𝑐) = (𝐹‘(𝑎 ↾ 𝑐)))}
14	1, 13	nfcxfr 2252	1 ⊢ Ⅎ𝑥recs(𝐹)

Syntax hints:   ∧ wa 103   = wceq 1314  {cab 2101  Ⅎwnfc 2242  ∀wral 2390  ∃wrex 2391  ∪ cuni 3702  Oncon0 4245   ↾ cres 4501   Fn wfn 5076  ‘cfv 5081  recscrecs 6155

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097

This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ral 2395  df-rex 2396  df-v 2659  df-un 3041  df-sn 3499  df-pr 3500  df-op 3502  df-uni 3703  df-br 3896  df-iota 5046  df-fv 5089  df-recs 6156

This theorem is referenced by:  nffrec  6247