Theorem nfres 5955

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 5653	. 2 ⊢ (𝐴 ↾ 𝐵) = (𝐴 ∩ (𝐵 × V))
2		nfres.1	. . 3 ⊢ Ⅎ𝑥𝐴
3		nfres.2	. . . 4 ⊢ Ⅎ𝑥𝐵
4		nfcv 2892	. . . 4 ⊢ Ⅎ𝑥V
5	3, 4	nfxp 5674	. . 3 ⊢ Ⅎ𝑥(𝐵 × V)
6	2, 5	nfin 4190	. 2 ⊢ Ⅎ𝑥(𝐴 ∩ (𝐵 × V))
7	1, 6	nfcxfr 2890	1 ⊢ Ⅎ𝑥(𝐴 ↾ 𝐵)

Syntax hints:  Ⅎwnfc 2877  Vcvv 3450   ∩ cin 3916   × cxp 5639   ↾ cres 5643

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-v 3452  df-in 3924  df-opab 5173  df-xp 5647  df-res 5653

This theorem is referenced by:  nfima  6042  nffrecs  8265  frsucmpt  8409  frsucmptn  8410  nfoi  9474  prdsdsf  24262  prdsxmet  24264  limciun  25802  nosupbnd2  27635  noinfbnd2  27650  2ndresdju  32580  gsumpart  33004  bnj1446  35042  bnj1447  35043  bnj1448  35044  bnj1466  35050  bnj1467  35051  bnj1519  35062  bnj1520  35063  bnj1529  35067  feqresmptf  45232  limcperiod  45633  xlimconst2  45840  cncfiooicclem1  45898  stoweidlem28  46033  nfdfat  47132  setrec2lem2  49687  setrec2  49688