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Theorem 1st2ndbr 6618
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 6615 . . 3  |-  ( ( Rel  B  /\  A  e.  B )  ->  A  =  <. ( 1st `  A
) ,  ( 2nd `  A ) >. )
2 simpr 461 . . 3  |-  ( ( Rel  B  /\  A  e.  B )  ->  A  e.  B )
31, 2eqeltrrd 2513 . 2  |-  ( ( Rel  B  /\  A  e.  B )  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  B )
4 df-br 4288 . 2  |-  ( ( 1st `  A ) B ( 2nd `  A
)  <->  <. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  B
)
53, 4sylibr 212 1  |-  ( ( Rel  B  /\  A  e.  B )  ->  ( 1st `  A ) B ( 2nd `  A
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    e. wcel 1756   <.cop 3878   class class class wbr 4287   Rel wrel 4840   ` cfv 5413   1stc1st 6570   2ndc2nd 6571
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-rab 2719  df-v 2969  df-sbc 3182  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-iota 5376  df-fun 5415  df-fv 5421  df-1st 6572  df-2nd 6573
This theorem is referenced by:  cofuval  14784  cofu1  14786  cofu2  14788  cofucl  14790  cofuass  14791  cofulid  14792  cofurid  14793  funcres  14798  cofull  14836  cofth  14837  isnat2  14850  fuccocl  14866  fucidcl  14867  fuclid  14868  fucrid  14869  fucass  14870  fucsect  14874  fucinv  14875  invfuc  14876  fuciso  14877  natpropd  14878  fucpropd  14879  homahom  14899  homadm  14900  homacd  14901  homadmcd  14902  catciso  14967  prfval  15001  prfcl  15005  prf1st  15006  prf2nd  15007  1st2ndprf  15008  evlfcllem  15023  evlfcl  15024  curf1cl  15030  curf2cl  15033  curfcl  15034  uncf1  15038  uncf2  15039  curfuncf  15040  uncfcurf  15041  diag1cl  15044  diag2cl  15048  curf2ndf  15049  yon1cl  15065  oyon1cl  15073  yonedalem1  15074  yonedalem21  15075  yonedalem3a  15076  yonedalem4c  15079  yonedalem22  15080  yonedalem3b  15081  yonedalem3  15082  yonedainv  15083  yonffthlem  15084  yoniso  15087  utop2nei  19800  utop3cls  19801
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