Theorem nfres 6011

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 5712	. 2 ⊢ (𝐴 ↾ 𝐵) = (𝐴 ∩ (𝐵 × V))
2		nfres.1	. . 3 ⊢ Ⅎ𝑥𝐴
3		nfres.2	. . . 4 ⊢ Ⅎ𝑥𝐵
4		nfcv 2908	. . . 4 ⊢ Ⅎ𝑥V
5	3, 4	nfxp 5733	. . 3 ⊢ Ⅎ𝑥(𝐵 × V)
6	2, 5	nfin 4245	. 2 ⊢ Ⅎ𝑥(𝐴 ∩ (𝐵 × V))
7	1, 6	nfcxfr 2906	1 ⊢ Ⅎ𝑥(𝐴 ↾ 𝐵)

Syntax hints:  Ⅎwnfc 2893  Vcvv 3488   ∩ cin 3975   × cxp 5698   ↾ cres 5702

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-tru 1540  df-ex 1778  df-nf 1782  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-v 3490  df-in 3983  df-opab 5229  df-xp 5706  df-res 5712

This theorem is referenced by:  nfima  6097  nffrecs  8324  nfwrecsOLD  8358  frsucmpt  8494  frsucmptn  8495  nfoi  9583  prdsdsf  24398  prdsxmet  24400  limciun  25949  nosupbnd2  27779  noinfbnd2  27794  2ndresdju  32667  gsumpart  33038  bnj1446  35021  bnj1447  35022  bnj1448  35023  bnj1466  35029  bnj1467  35030  bnj1519  35041  bnj1520  35042  bnj1529  35046  feqresmptf  45138  limcperiod  45549  xlimconst2  45756  cncfiooicclem1  45814  stoweidlem28  45949  nfdfat  47042  setrec2lem2  48786  setrec2  48787