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Theorem opres 4911
Description: Ordered pair membership in a restriction when the first member belongs to the restricting class. (Contributed by NM, 30-Apr-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Hypothesis
Ref Expression
opres.1  |-  B  e. 
_V
Assertion
Ref Expression
opres  |-  ( A  e.  D  ->  ( <. A ,  B >.  e.  ( C  |`  D )  <->  <. A ,  B >.  e.  C ) )

Proof of Theorem opres
StepHypRef Expression
1 opres.1 . . 3  |-  B  e. 
_V
21opelres 4907 . 2  |-  ( <. A ,  B >.  e.  ( C  |`  D )  <-> 
( <. A ,  B >.  e.  C  /\  A  e.  D ) )
32rbaib 921 1  |-  ( A  e.  D  ->  ( <. A ,  B >.  e.  ( C  |`  D )  <->  <. A ,  B >.  e.  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 105    e. wcel 2148   _Vcvv 2737   <.cop 3594    |` cres 4624
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4205
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-opab 4062  df-xp 4628  df-res 4634
This theorem is referenced by:  resieq  4912  2elresin  5322
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