Theorem res0 5009

Description: A restriction to the empty set is empty. (Contributed by NM, 12-Nov-1994.)

Assertion

Ref	Expression
res0	|- ( A |` (/) ) = (/)

Proof of Theorem res0

Step	Hyp	Ref	Expression
1		df-res 4731	. 2 |- ( A |` (/) ) = ( A i^i ( (/) X. _V ) )
2		0xp 4799	. . 3 |- ( (/) X. _V ) = (/)
3	2	ineq2i 3402	. 2 |- ( A i^i ( (/) X. _V ) ) = ( A i^i (/) )
4		in0 3526	. 2 |- ( A i^i (/) ) = (/)
5	1, 3, 4	3eqtri 2254	1 |- ( A |` (/) ) = (/)

Syntax hints:   = wceq 1395   _Vcvv 2799   i^i cin 3196   (/)c0 3491   X. cxp 4717   |` cres 4721

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-opab 4146  df-xp 4725  df-res 4731

This theorem is referenced by:  ima0  5087  resdisj  5157  smo0  6444  tfr0dm  6468  tfr0  6469  fnfi  7103  setsslid  13083