Theorem nfres 5430

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 5155	. 2 ⊢ (𝐴 ↾ 𝐵) = (𝐴 ∩ (𝐵 × V))
2		nfres.1	. . 3 ⊢ Ⅎ𝑥𝐴
3		nfres.2	. . . 4 ⊢ Ⅎ𝑥𝐵
4		nfcv 2793	. . . 4 ⊢ Ⅎ𝑥V
5	3, 4	nfxp 5176	. . 3 ⊢ Ⅎ𝑥(𝐵 × V)
6	2, 5	nfin 3853	. 2 ⊢ Ⅎ𝑥(𝐴 ∩ (𝐵 × V))
7	1, 6	nfcxfr 2791	1 ⊢ Ⅎ𝑥(𝐴 ↾ 𝐵)

Syntax hints:  Ⅎwnfc 2780  Vcvv 3231   ∩ cin 3606   × cxp 5141   ↾ cres 5145

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-rab 2950  df-in 3614  df-opab 4746  df-xp 5149  df-res 5155

This theorem is referenced by:  nfima  5509  nfwrecs  7454  frsucmpt  7578  frsucmptn  7579  nfoi  8460  prdsdsf  22219  prdsxmet  22221  limciun  23703  bnj1446  31239  bnj1447  31240  bnj1448  31241  bnj1466  31247  bnj1467  31248  bnj1519  31259  bnj1520  31260  bnj1529  31264  trpredlem1  31851  trpredrec  31862  nffrecs  31903  nosupbnd2  31987  wessf1ornlem  39685  feqresmptf  39747  limcperiod  40178  xlimconst2  40379  cncfiooicclem1  40424  stoweidlem28  40563  nfdfat  41531  setrec2lem2  42766  setrec2  42767