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Theorem 1st2nd 6009
Description: Reconstruction of a member of a relation in terms of its ordered pair components. (Contributed by NM, 29-Aug-2006.)
Assertion
Ref Expression
1st2nd  |-  ( ( Rel  B  /\  A  e.  B )  ->  A  =  <. ( 1st `  A
) ,  ( 2nd `  A ) >. )

Proof of Theorem 1st2nd
StepHypRef Expression
1 df-rel 4484 . . 3  |-  ( Rel 
B  <->  B  C_  ( _V 
X.  _V ) )
2 ssel2 3042 . . 3  |-  ( ( B  C_  ( _V  X.  _V )  /\  A  e.  B )  ->  A  e.  ( _V  X.  _V ) )
31, 2sylanb 280 . 2  |-  ( ( Rel  B  /\  A  e.  B )  ->  A  e.  ( _V  X.  _V ) )
4 1st2nd2 6003 . 2  |-  ( A  e.  ( _V  X.  _V )  ->  A  = 
<. ( 1st `  A
) ,  ( 2nd `  A ) >. )
53, 4syl 14 1  |-  ( ( Rel  B  /\  A  e.  B )  ->  A  =  <. ( 1st `  A
) ,  ( 2nd `  A ) >. )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1299    e. wcel 1448   _Vcvv 2641    C_ wss 3021   <.cop 3477    X. cxp 4475   Rel wrel 4482   ` cfv 5059   1stc1st 5967   2ndc2nd 5968
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-pow 4038  ax-pr 4069  ax-un 4293
This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-rex 2381  df-v 2643  df-sbc 2863  df-un 3025  df-in 3027  df-ss 3034  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-br 3876  df-opab 3930  df-mpt 3931  df-id 4153  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-iota 5024  df-fun 5061  df-fv 5067  df-1st 5969  df-2nd 5970
This theorem is referenced by:  2ndrn  6011  1st2ndbr  6012  elopabi  6023  cnvf1olem  6051  fsumcnv  11045
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