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Theorem 1st2ndbr 6185
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 6182 . . 3  |-  ( ( Rel  B  /\  A  e.  B )  ->  A  =  <. ( 1st `  A
) ,  ( 2nd `  A ) >. )
2 simpr 447 . . 3  |-  ( ( Rel  B  /\  A  e.  B )  ->  A  e.  B )
31, 2eqeltrrd 2371 . 2  |-  ( ( Rel  B  /\  A  e.  B )  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  B )
4 df-br 4040 . 2  |-  ( ( 1st `  A ) B ( 2nd `  A
)  <->  <. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  B
)
53, 4sylibr 203 1  |-  ( ( Rel  B  /\  A  e.  B )  ->  ( 1st `  A ) B ( 2nd `  A
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    e. wcel 1696   <.cop 3656   class class class wbr 4039   Rel wrel 4710   ` cfv 5271   1stc1st 6136   2ndc2nd 6137
This theorem is referenced by:  cofuval  13772  cofu1  13774  cofu2  13776  cofucl  13778  cofuass  13779  cofulid  13780  cofurid  13781  funcres  13786  cofull  13824  cofth  13825  isnat2  13838  fuccocl  13854  fucidcl  13855  fuclid  13856  fucrid  13857  fucass  13858  fucsect  13862  fucinv  13863  invfuc  13864  fuciso  13865  natpropd  13866  fucpropd  13867  homahom  13887  homadm  13888  homacd  13889  homadmcd  13890  catciso  13955  prfval  13989  prfcl  13993  prf1st  13994  prf2nd  13995  1st2ndprf  13996  evlfcllem  14011  evlfcl  14012  curf1cl  14018  curf2cl  14021  curfcl  14022  uncf1  14026  uncf2  14027  curfuncf  14028  uncfcurf  14029  diag1cl  14032  diag2cl  14036  curf2ndf  14037  yon1cl  14053  oyon1cl  14061  yonedalem1  14062  yonedalem21  14063  yonedalem3a  14064  yonedalem4c  14067  yonedalem22  14068  yonedalem3b  14069  yonedalem3  14070  yonedainv  14071  yonffthlem  14072  yoniso  14075
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-iota 5235  df-fun 5273  df-fv 5279  df-1st 6138  df-2nd 6139
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