Theorem nfres 5398

Description: Bound-variable hypothesis builder for restriction. (Contributed by NM, 15-Sep-2003.) (Revised by David Abernethy, 19-Jun-2012.)

Hypotheses

Ref	Expression
nfres.1	⊢ Ⅎ𝑥𝐴
nfres.2	⊢ Ⅎ𝑥𝐵

Assertion

Ref	Expression
nfres	⊢ Ⅎ𝑥(𝐴 ↾ 𝐵)

Proof of Theorem nfres

Step	Hyp	Ref	Expression
1		df-res 5126	. 2 ⊢ (𝐴 ↾ 𝐵) = (𝐴 ∩ (𝐵 × V))
2		nfres.1	. . 3 ⊢ Ⅎ𝑥𝐴
3		nfres.2	. . . 4 ⊢ Ⅎ𝑥𝐵
4		nfcv 2764	. . . 4 ⊢ Ⅎ𝑥V
5	3, 4	nfxp 5142	. . 3 ⊢ Ⅎ𝑥(𝐵 × V)
6	2, 5	nfin 3820	. 2 ⊢ Ⅎ𝑥(𝐴 ∩ (𝐵 × V))
7	1, 6	nfcxfr 2762	1 ⊢ Ⅎ𝑥(𝐴 ↾ 𝐵)

Syntax hints:  Ⅎwnfc 2751  Vcvv 3200   ∩ cin 3573   × cxp 5112   ↾ cres 5116

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-in 3581  df-opab 4713  df-xp 5120  df-res 5126

This theorem is referenced by:  nfima  5474  nfwrecs  7409  frsucmpt  7533  frsucmptn  7534  nfoi  8419  prdsdsf  22172  prdsxmet  22174  limciun  23658  bnj1446  31113  bnj1447  31114  bnj1448  31115  bnj1466  31121  bnj1467  31122  bnj1519  31133  bnj1520  31134  bnj1529  31138  trpredlem1  31727  trpredrec  31738  nosupbnd2  31862  wessf1ornlem  39371  feqresmptf  39433  limcperiod  39860  xlimconst2  40061  cncfiooicclem1  40106  stoweidlem28  40245  nfdfat  41210  setrec2lem2  42441  setrec2  42442