Theorem nfres 5893

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 5601	. 2 ⊢ (𝐴 ↾ 𝐵) = (𝐴 ∩ (𝐵 × V))
2		nfres.1	. . 3 ⊢ Ⅎ𝑥𝐴
3		nfres.2	. . . 4 ⊢ Ⅎ𝑥𝐵
4		nfcv 2907	. . . 4 ⊢ Ⅎ𝑥V
5	3, 4	nfxp 5622	. . 3 ⊢ Ⅎ𝑥(𝐵 × V)
6	2, 5	nfin 4150	. 2 ⊢ Ⅎ𝑥(𝐴 ∩ (𝐵 × V))
7	1, 6	nfcxfr 2905	1 ⊢ Ⅎ𝑥(𝐴 ↾ 𝐵)

Syntax hints:  Ⅎwnfc 2887  Vcvv 3432   ∩ cin 3886   × cxp 5587   ↾ cres 5591

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-rab 3073  df-in 3894  df-opab 5137  df-xp 5595  df-res 5601

This theorem is referenced by:  nfima  5977  nffrecs  8099  nfwrecsOLD  8133  frsucmpt  8269  frsucmptn  8270  nfoi  9273  prdsdsf  23520  prdsxmet  23522  limciun  25058  2ndresdju  30986  gsumpart  31315  bnj1446  33025  bnj1447  33026  bnj1448  33027  bnj1466  33033  bnj1467  33034  bnj1519  33045  bnj1520  33046  bnj1529  33050  nosupbnd2  33919  noinfbnd2  33934  feqresmptf  42772  limcperiod  43169  xlimconst2  43376  cncfiooicclem1  43434  stoweidlem28  43569  nfdfat  44619  setrec2lem2  46400  setrec2  46401