Theorem nfres 6001

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 5700	. 2 ⊢ (𝐴 ↾ 𝐵) = (𝐴 ∩ (𝐵 × V))
2		nfres.1	. . 3 ⊢ Ⅎ𝑥𝐴
3		nfres.2	. . . 4 ⊢ Ⅎ𝑥𝐵
4		nfcv 2902	. . . 4 ⊢ Ⅎ𝑥V
5	3, 4	nfxp 5721	. . 3 ⊢ Ⅎ𝑥(𝐵 × V)
6	2, 5	nfin 4231	. 2 ⊢ Ⅎ𝑥(𝐴 ∩ (𝐵 × V))
7	1, 6	nfcxfr 2900	1 ⊢ Ⅎ𝑥(𝐴 ↾ 𝐵)

Syntax hints:  Ⅎwnfc 2887  Vcvv 3477   ∩ cin 3961   × cxp 5686   ↾ cres 5690

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1539  df-ex 1776  df-nf 1780  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-v 3479  df-in 3969  df-opab 5210  df-xp 5694  df-res 5700

This theorem is referenced by:  nfima  6087  nffrecs  8306  nfwrecsOLD  8340  frsucmpt  8476  frsucmptn  8477  nfoi  9551  prdsdsf  24392  prdsxmet  24394  limciun  25943  nosupbnd2  27775  noinfbnd2  27790  2ndresdju  32665  gsumpart  33042  bnj1446  35037  bnj1447  35038  bnj1448  35039  bnj1466  35045  bnj1467  35046  bnj1519  35057  bnj1520  35058  bnj1529  35062  feqresmptf  45173  limcperiod  45583  xlimconst2  45790  cncfiooicclem1  45848  stoweidlem28  45983  nfdfat  47076  setrec2lem2  48924  setrec2  48925