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Theorem opelresg 4961
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 ) ) )
Dummy variable  y is distinct from all other variables.

Proof of Theorem opelresg
StepHypRef Expression
1 opeq2 3798 . . 3  |-  ( y  =  B  ->  <. A , 
y >.  =  <. A ,  B >. )
21eleq1d 2350 . 2  |-  ( y  =  B  ->  ( <. A ,  y >.  e.  ( C  |`  D )  <->  <. A ,  B >.  e.  ( C  |`  D ) ) )
31eleq1d 2350 . . 3  |-  ( y  =  B  ->  ( <. A ,  y >.  e.  C  <->  <. A ,  B >.  e.  C ) )
43anbi1d 687 . 2  |-  ( y  =  B  ->  (
( <. A ,  y
>.  e.  C  /\  A  e.  D )  <->  ( <. A ,  B >.  e.  C  /\  A  e.  D
) ) )
5 vex 2792 . . 3  |-  y  e. 
_V
65opelres 4959 . 2  |-  ( <. A ,  y >.  e.  ( C  |`  D )  <-> 
( <. A ,  y
>.  e.  C  /\  A  e.  D ) )
72, 4, 6vtoclbg 2845 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 6    <-> wb 178    /\ wa 360    = wceq 1624    e. wcel 1685   <.cop 3644    |` cres 4690
This theorem is referenced by:  brresg  4962  opelresiOLD  4965  opelresi  4966  issref  5055
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pr 4213
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-sn 3647  df-pr 3648  df-op 3650  df-opab 4079  df-xp 4694  df-res 4700
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