Theorem opelresg 5008

Description: Ordered pair membership in a restriction. Exercise 13 of [TakeutiZaring] p. 25. (Contributed by NM, 14-Oct-2005.)

Assertion

Ref	Expression
opelresg	|- ( B e. V -> ( <. A , B >. e. ( C |` D ) <-> ( <. A , B >. e. C /\ A e. D ) ) )

Proof of Theorem opelresg

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		opeq2 3857	. . 3 |- ( y = B -> <. A , y >. = <. A , B >. )
2	1	eleq1d 2298	. 2 |- ( y = B -> ( <. A , y >. e. ( C |` D ) <-> <. A , B >. e. ( C |` D ) ) )
3	1	eleq1d 2298	. . 3 |- ( y = B -> ( <. A , y >. e. C <-> <. A , B >. e. C ) )
4	3	anbi1d 465	. 2 |- ( y = B -> ( ( <. A , y >. e. C /\ A e. D ) <-> ( <. A , B >. e. C /\ A e. D ) ) )
5		vex 2802	. . 3 |- y e. _V
6	5	opelres 5006	. 2 |- ( <. A , y >. e. ( C |` D ) <-> ( <. A , y >. e. C /\ A e. D ) )
7	2, 4, 6	vtoclbg 2862	1 |- ( B e. V -> ( <. A , B >. e. ( C |` D ) <-> ( <. A , B >. e. C /\ A e. D ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   e. wcel 2200   <.cop 3669   |` cres 4718

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 4201  ax-pow 4257  ax-pr 4292

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 4145  df-xp 4722  df-res 4728

This theorem is referenced by:  brresg  5009  opelresi  5012  issref  5107