Theorem opres 4951

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 4947	. 2 |- ( <. A , B >. e. ( C |` D ) <-> ( <. A , B >. e. C /\ A e. D ) )
3	2	rbaib 922	1 |- ( A e. D -> ( <. A , B >. e. ( C |` D ) <-> <. A , B >. e. C ) )

Syntax hints:   -> wi 4   <-> wb 105   e. wcel 2164   _Vcvv 2760   <.cop 3621   |` cres 4661

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-opab 4091  df-xp 4665  df-res 4671

This theorem is referenced by:  resieq  4952  2elresin  5365