Theorem nfres 5984

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 5689	. 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 5710	. . 3 ⊢ Ⅎ𝑥(𝐵 × V)
6	2, 5	nfin 4217	. 2 ⊢ Ⅎ𝑥(𝐴 ∩ (𝐵 × V))
7	1, 6	nfcxfr 2902	1 ⊢ Ⅎ𝑥(𝐴 ↾ 𝐵)

Syntax hints:  Ⅎwnfc 2884  Vcvv 3475   ∩ cin 3948   × cxp 5675   ↾ cres 5679

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 3956  df-opab 5212  df-xp 5683  df-res 5689

This theorem is referenced by:  nfima  6068  nffrecs  8268  nfwrecsOLD  8302  frsucmpt  8438  frsucmptn  8439  nfoi  9509  prdsdsf  23873  prdsxmet  23875  limciun  25411  nosupbnd2  27219  noinfbnd2  27234  2ndresdju  31874  gsumpart  32207  bnj1446  34056  bnj1447  34057  bnj1448  34058  bnj1466  34064  bnj1467  34065  bnj1519  34076  bnj1520  34077  bnj1529  34081  feqresmptf  43931  limcperiod  44344  xlimconst2  44551  cncfiooicclem1  44609  stoweidlem28  44744  nfdfat  45835  setrec2lem2  47739  setrec2  47740