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Theorem 1st2ndbr 5892
Description: Express an element of a relation as a relationship between first and second components. (Contributed by Mario Carneiro, 22-Jun-2016.)
Assertion
Ref Expression
1st2ndbr  |-  ( ( Rel  B  /\  A  e.  B )  ->  ( 1st `  A ) B ( 2nd `  A
) )

Proof of Theorem 1st2ndbr
StepHypRef Expression
1 1st2nd 5889 . . 3  |-  ( ( Rel  B  /\  A  e.  B )  ->  A  =  <. ( 1st `  A
) ,  ( 2nd `  A ) >. )
2 simpr 108 . . 3  |-  ( ( Rel  B  /\  A  e.  B )  ->  A  e.  B )
31, 2eqeltrrd 2162 . 2  |-  ( ( Rel  B  /\  A  e.  B )  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  B )
4 df-br 3815 . 2  |-  ( ( 1st `  A ) B ( 2nd `  A
)  <->  <. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  B
)
53, 4sylibr 132 1  |-  ( ( Rel  B  /\  A  e.  B )  ->  ( 1st `  A ) B ( 2nd `  A
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    e. wcel 1436   <.cop 3428   class class class wbr 3814   Rel wrel 4409   ` cfv 4972   1stc1st 5847   2ndc2nd 5848
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 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3925  ax-pow 3977  ax-pr 4003  ax-un 4227
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-v 2616  df-sbc 2829  df-un 2990  df-in 2992  df-ss 2999  df-pw 3411  df-sn 3431  df-pr 3432  df-op 3434  df-uni 3631  df-br 3815  df-opab 3869  df-mpt 3870  df-id 4087  df-xp 4410  df-rel 4411  df-cnv 4412  df-co 4413  df-dm 4414  df-rn 4415  df-iota 4937  df-fun 4974  df-fv 4980  df-1st 5849  df-2nd 5850
This theorem is referenced by: (None)
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