Theorem nfres 5941

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 5637	. 2 ⊢ (𝐴 ↾ 𝐵) = (𝐴 ∩ (𝐵 × V))
2		nfres.1	. . 3 ⊢ Ⅎ𝑥𝐴
3		nfres.2	. . . 4 ⊢ Ⅎ𝑥𝐵
4		nfcv 2899	. . . 4 ⊢ Ⅎ𝑥V
5	3, 4	nfxp 5658	. . 3 ⊢ Ⅎ𝑥(𝐵 × V)
6	2, 5	nfin 4177	. 2 ⊢ Ⅎ𝑥(𝐴 ∩ (𝐵 × V))
7	1, 6	nfcxfr 2897	1 ⊢ Ⅎ𝑥(𝐴 ↾ 𝐵)

Syntax hints:  Ⅎwnfc 2884  Vcvv 3441   ∩ cin 3901   × cxp 5623   ↾ cres 5627

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-ex 1782  df-nf 1786  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-v 3443  df-in 3909  df-opab 5162  df-xp 5631  df-res 5637

This theorem is referenced by:  nfima  6028  nffrecs  8227  frsucmpt  8371  frsucmptn  8372  nfoi  9423  prdsdsf  24315  prdsxmet  24317  limciun  25855  nosupbnd2  27688  noinfbnd2  27703  2ndresdju  32709  gsumpart  33127  bnj1446  35182  bnj1447  35183  bnj1448  35184  bnj1466  35190  bnj1467  35191  bnj1519  35202  bnj1520  35203  bnj1529  35207  feqresmptf  45511  limcperiod  45910  xlimconst2  46115  cncfiooicclem1  46173  stoweidlem28  46308  nfdfat  47409  setrec2lem2  49975  setrec2  49976