Theorem opres 5022

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 5018	. 2 |- ( <. A , B >. e. ( C |` D ) <-> ( <. A , B >. e. C /\ A e. D ) )
3	2	rbaib 928	1 |- ( A e. D -> ( <. A , B >. e. ( C |` D ) <-> <. A , B >. e. C ) )

Syntax hints:   -> wi 4   <-> wb 105   e. wcel 2202   _Vcvv 2802   <.cop 3672   |` cres 4727

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-opab 4151  df-xp 4731  df-res 4737

This theorem is referenced by:  resieq  5023  2elresin  5443