Theorem opres 5014

Description: Ordered pair membership in a restriction when the first member belongs to the restricting class. (Contributed by NM, 30-Apr-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Hypothesis

Ref	Expression
opres.1	|- B e. _V

Assertion

Ref	Expression
opres	|- ( A e. D -> ( <. A , B >. e. ( C |` D ) <-> <. A , B >. e. C ) )

Proof of Theorem opres

Step	Hyp	Ref	Expression
1		opres.1	. . 3 |- B e. _V
2	1	opelres 5010	. 2 |- ( <. A , B >. e. ( C |` D ) <-> ( <. A , B >. e. C /\ A e. D ) )
3	2	rbaib 926	1 |- ( A e. D -> ( <. A , B >. e. ( C |` D ) <-> <. A , B >. e. C ) )

Syntax hints:   -> wi 4   <-> wb 105   e. wcel 2200   _Vcvv 2799   <.cop 3669   |` 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-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-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  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:  resieq  5015  2elresin  5434