Theorem nfres 5948

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 5644	. 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 5665	. . 3 ⊢ Ⅎ𝑥(𝐵 × V)
6	2, 5	nfin 4178	. 2 ⊢ Ⅎ𝑥(𝐴 ∩ (𝐵 × V))
7	1, 6	nfcxfr 2897	1 ⊢ Ⅎ𝑥(𝐴 ↾ 𝐵)

Syntax hints:  Ⅎwnfc 2884  Vcvv 3442   ∩ cin 3902   × cxp 5630   ↾ cres 5634

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 3444  df-in 3910  df-opab 5163  df-xp 5638  df-res 5644

This theorem is referenced by:  nfima  6035  nffrecs  8235  frsucmpt  8379  frsucmptn  8380  nfoi  9431  prdsdsf  24323  prdsxmet  24325  limciun  25863  nosupbnd2  27696  noinfbnd2  27711  2ndresdju  32738  gsumpart  33156  bnj1446  35220  bnj1447  35221  bnj1448  35222  bnj1466  35228  bnj1467  35229  bnj1519  35240  bnj1520  35241  bnj1529  35245  feqresmptf  45586  limcperiod  45985  xlimconst2  46190  cncfiooicclem1  46248  stoweidlem28  46383  nfdfat  47484  setrec2lem2  50050  setrec2  50051