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Theorem opelresg 4962
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 3797 . . 3  |-  ( y  =  B  ->  <. A , 
y >.  =  <. A ,  B >. )
21eleq1d 2349 . 2  |-  ( y  =  B  ->  ( <. A ,  y >.  e.  ( C  |`  D )  <->  <. A ,  B >.  e.  ( C  |`  D ) ) )
31eleq1d 2349 . . 3  |-  ( y  =  B  ->  ( <. A ,  y >.  e.  C  <->  <. A ,  B >.  e.  C ) )
43anbi1d 685 . 2  |-  ( y  =  B  ->  (
( <. A ,  y
>.  e.  C  /\  A  e.  D )  <->  ( <. A ,  B >.  e.  C  /\  A  e.  D
) ) )
5 vex 2791 . . 3  |-  y  e. 
_V
65opelres 4960 . 2  |-  ( <. A ,  y >.  e.  ( C  |`  D )  <-> 
( <. A ,  y
>.  e.  C  /\  A  e.  D ) )
72, 4, 6vtoclbg 2844 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    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   <.cop 3643    |` cres 4691
This theorem is referenced by:  brresg  4963  opelresiOLD  4966  opelresi  4967  issref  5056
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-opab 4078  df-xp 4695  df-res 4701
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