ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  opelresg Unicode version

Theorem opelresg 4641
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 3573 . . 3  |-  ( y  =  B  ->  <. A , 
y >.  =  <. A ,  B >. )
21eleq1d 2148 . 2  |-  ( y  =  B  ->  ( <. A ,  y >.  e.  ( C  |`  D )  <->  <. A ,  B >.  e.  ( C  |`  D ) ) )
31eleq1d 2148 . . 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 2605 . . 3  |-  y  e. 
_V
65opelres 4639 . 2  |-  ( <. A ,  y >.  e.  ( C  |`  D )  <-> 
( <. A ,  y
>.  e.  C  /\  A  e.  D ) )
72, 4, 6vtoclbg 2660 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 1285    e. wcel 1434   <.cop 3403    |` cres 4367
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3898  ax-pow 3950  ax-pr 3966
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-opab 3842  df-xp 4371  df-res 4377
This theorem is referenced by:  brresg  4642  opelresi  4645  issref  4731
  Copyright terms: Public domain W3C validator