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Theorem opelresg 5145
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 3977 . . 3  |-  ( y  =  B  ->  <. A , 
y >.  =  <. A ,  B >. )
21eleq1d 2501 . 2  |-  ( y  =  B  ->  ( <. A ,  y >.  e.  ( C  |`  D )  <->  <. A ,  B >.  e.  ( C  |`  D ) ) )
31eleq1d 2501 . . 3  |-  ( y  =  B  ->  ( <. A ,  y >.  e.  C  <->  <. A ,  B >.  e.  C ) )
43anbi1d 686 . 2  |-  ( y  =  B  ->  (
( <. A ,  y
>.  e.  C  /\  A  e.  D )  <->  ( <. A ,  B >.  e.  C  /\  A  e.  D
) ) )
5 vex 2951 . . 3  |-  y  e. 
_V
65opelres 5143 . 2  |-  ( <. A ,  y >.  e.  ( C  |`  D )  <-> 
( <. A ,  y
>.  e.  C  /\  A  e.  D ) )
72, 4, 6vtoclbg 3004 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 177    /\ wa 359    = wceq 1652    e. wcel 1725   <.cop 3809    |` cres 4872
This theorem is referenced by:  brresg  5146  opelresiOLD  5149  opelresi  5150  issref  5239
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-opab 4259  df-xp 4876  df-res 4882
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