Theorem nfres 5980

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 5686	. 2 ⊢ (𝐴 ↾ 𝐵) = (𝐴 ∩ (𝐵 × V))
2		nfres.1	. . 3 ⊢ Ⅎ𝑥𝐴
3		nfres.2	. . . 4 ⊢ Ⅎ𝑥𝐵
4		nfcv 2904	. . . 4 ⊢ Ⅎ𝑥V
5	3, 4	nfxp 5707	. . 3 ⊢ Ⅎ𝑥(𝐵 × V)
6	2, 5	nfin 4214	. 2 ⊢ Ⅎ𝑥(𝐴 ∩ (𝐵 × V))
7	1, 6	nfcxfr 2902	1 ⊢ Ⅎ𝑥(𝐴 ↾ 𝐵)

Syntax hints:  Ⅎwnfc 2884  Vcvv 3475   ∩ cin 3945   × cxp 5672   ↾ cres 5676

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-rab 3434  df-in 3953  df-opab 5209  df-xp 5680  df-res 5686

This theorem is referenced by:  nfima  6064  nffrecs  8262  nfwrecsOLD  8296  frsucmpt  8432  frsucmptn  8433  nfoi  9504  prdsdsf  23854  prdsxmet  23856  limciun  25392  nosupbnd2  27198  noinfbnd2  27213  2ndresdju  31851  gsumpart  32184  bnj1446  33993  bnj1447  33994  bnj1448  33995  bnj1466  34001  bnj1467  34002  bnj1519  34013  bnj1520  34014  bnj1529  34018  feqresmptf  43864  limcperiod  44278  xlimconst2  44485  cncfiooicclem1  44543  stoweidlem28  44678  nfdfat  45769  setrec2lem2  47640  setrec2  47641