Theorem resex 4987

Description: The restriction of a set is a set. (Contributed by Jeff Madsen, 19-Jun-2011.)

Hypothesis

Ref	Expression
resex.1	|- A e. _V

Assertion

Ref	Expression
resex	|- ( A |` B ) e. _V

Proof of Theorem resex

Step	Hyp	Ref	Expression
1		resex.1	. 2 |- A e. _V
2		resexg 4986	. 2 |- ( A e. _V -> ( A |` B ) e. _V )
3	1, 2	ax-mp 5	1 |- ( A |` B ) e. _V

Syntax hints:   e. wcel 2167   _Vcvv 2763   |` cres 4665

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-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178  ax-sep 4151

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-in 3163  df-ss 3170  df-res 4675

This theorem is referenced by:  sbthlemi10  7032  finomni  7206  ctinf  12647  znval  14192