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Theorem 1st2ndbr 6398
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 6395 . . 3  |-  ( ( Rel  B  /\  A  e.  B )  ->  A  =  <. ( 1st `  A
) ,  ( 2nd `  A ) >. )
2 simpr 449 . . 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 4215 . 2  |-  ( ( 1st `  A ) B ( 2nd `  A
)  <->  <. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  B
)
53, 4sylibr 205 1  |-  ( ( Rel  B  /\  A  e.  B )  ->  ( 1st `  A ) B ( 2nd `  A
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    e. wcel 1726   <.cop 3819   class class class wbr 4214   Rel wrel 4885   ` cfv 5456   1stc1st 6349   2ndc2nd 6350
This theorem is referenced by:  cofuval  14081  cofu1  14083  cofu2  14085  cofucl  14087  cofuass  14088  cofulid  14089  cofurid  14090  funcres  14095  cofull  14133  cofth  14134  isnat2  14147  fuccocl  14163  fucidcl  14164  fuclid  14165  fucrid  14166  fucass  14167  fucsect  14171  fucinv  14172  invfuc  14173  fuciso  14174  natpropd  14175  fucpropd  14176  homahom  14196  homadm  14197  homacd  14198  homadmcd  14199  catciso  14264  prfval  14298  prfcl  14302  prf1st  14303  prf2nd  14304  1st2ndprf  14305  evlfcllem  14320  evlfcl  14321  curf1cl  14327  curf2cl  14330  curfcl  14331  uncf1  14335  uncf2  14336  curfuncf  14337  uncfcurf  14338  diag1cl  14341  diag2cl  14345  curf2ndf  14346  yon1cl  14362  oyon1cl  14370  yonedalem1  14371  yonedalem21  14372  yonedalem3a  14373  yonedalem4c  14376  yonedalem22  14377  yonedalem3b  14378  yonedalem3  14379  yonedainv  14380  yonffthlem  14381  yoniso  14384  utop2nei  18282  utop3cls  18283
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-iota 5420  df-fun 5458  df-fv 5464  df-1st 6351  df-2nd 6352
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