Theorem nfres 5925

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 5623	. 2 ⊢ (𝐴 ↾ 𝐵) = (𝐴 ∩ (𝐵 × V))
2		nfres.1	. . 3 ⊢ Ⅎ𝑥𝐴
3		nfres.2	. . . 4 ⊢ Ⅎ𝑥𝐵
4		nfcv 2894	. . . 4 ⊢ Ⅎ𝑥V
5	3, 4	nfxp 5644	. . 3 ⊢ Ⅎ𝑥(𝐵 × V)
6	2, 5	nfin 4169	. 2 ⊢ Ⅎ𝑥(𝐴 ∩ (𝐵 × V))
7	1, 6	nfcxfr 2892	1 ⊢ Ⅎ𝑥(𝐴 ↾ 𝐵)

Syntax hints:  Ⅎwnfc 2879  Vcvv 3436   ∩ cin 3896   × cxp 5609   ↾ cres 5613

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-ex 1781  df-nf 1785  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-v 3438  df-in 3904  df-opab 5149  df-xp 5617  df-res 5623

This theorem is referenced by:  nfima  6012  nffrecs  8208  frsucmpt  8352  frsucmptn  8353  nfoi  9395  prdsdsf  24277  prdsxmet  24279  limciun  25817  nosupbnd2  27650  noinfbnd2  27665  2ndresdju  32623  gsumpart  33029  bnj1446  35049  bnj1447  35050  bnj1448  35051  bnj1466  35057  bnj1467  35058  bnj1519  35069  bnj1520  35070  bnj1529  35074  feqresmptf  45268  limcperiod  45668  xlimconst2  45873  cncfiooicclem1  45931  stoweidlem28  46066  nfdfat  47158  setrec2lem2  49726  setrec2  49727