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Theorem opelresg 4708
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.
StepHypRef Expression
1 opeq2 3618 . . 3  |-  ( y  =  B  ->  <. A , 
y >.  =  <. A ,  B >. )
21eleq1d 2156 . 2  |-  ( y  =  B  ->  ( <. A ,  y >.  e.  ( C  |`  D )  <->  <. A ,  B >.  e.  ( C  |`  D ) ) )
31eleq1d 2156 . . 3  |-  ( y  =  B  ->  ( <. A ,  y >.  e.  C  <->  <. A ,  B >.  e.  C ) )
43anbi1d 453 . 2  |-  ( y  =  B  ->  (
( <. A ,  y
>.  e.  C  /\  A  e.  D )  <->  ( <. A ,  B >.  e.  C  /\  A  e.  D
) ) )
5 vex 2622 . . 3  |-  y  e. 
_V
65opelres 4706 . 2  |-  ( <. A ,  y >.  e.  ( C  |`  D )  <-> 
( <. A ,  y
>.  e.  C  /\  A  e.  D ) )
72, 4, 6vtoclbg 2680 1  |-  ( B  e.  V  ->  ( <. A ,  B >.  e.  ( C  |`  D )  <-> 
( <. A ,  B >.  e.  C  /\  A  e.  D ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1289    e. wcel 1438   <.cop 3444    |` cres 4430
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-opab 3892  df-xp 4434  df-res 4440
This theorem is referenced by:  brresg  4709  opelresi  4712  issref  4801
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